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Paper XII: Patching Support, Hecke Packets, and Conditional Automorphic Completion
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515 symbol spellings in 1591 mathematical passages across 30 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
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=Definition passage · 325 passages
Search aliases: =
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
\AnnDefinition passage · 9 passages
Search aliases: Ann operatorname
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 580
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Ann_R(M)
\Ann_RDefinition passage · 9 passages
Search aliases: Ann_R Ann operatorname
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 580
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Ann_R(M)
\bar\rhoDefinition passage · 5 passages
Search aliases: barρ bar rho
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 210
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
\bar\rho
CDefinition passage · 50 passages
Search aliases: C
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 811
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
C
C_\inftyDefinition passage · 6 passages
Search aliases: C_∞ infty
Definition passage: Derived patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 863
[Derived patching datum] A derived patching datum consists of a Noetherian ring R_∞, an ideal a, a perfect complex C_∞ over R_∞, and identifications R≅ R_∞/ a, C≃ C_∞ mathbinmathop⊗limits_R_∞^ LR.
Exact mathematical passage
C_\infty
EDefinition passage · 54 passages
Search aliases: E
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1640
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
x:R\to E'
eDefinition passage · 100 passages
Search aliases: e
Definition passage: Derived completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1690
ndefinition[Derived completion datum at a point] A derived completion datum is defined in the same way, with a perfect patched complex C_∞ of full derived support and a finite level automorphic complex C ≃ C_∞ mathbinmathop⊗limits_R_∞^ LR in place of the ordinary modules. Under the Hecke ring isomorphism e:R≅ T, require an actual derived carrier identification between the scalar transport of this finite level complex and the automorphic Hecke complex. This identification retains the R action ind…
Exact mathematical passage
e:R\cong\TT
E'Definition passage · 46 passages
Search aliases: E'
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1640
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
x:R\to E'
FDefinition passage · 11 passages
Search aliases: F
Definition passage: Algebraic Frobenius spine · Read passage · Paper_XII_Traditional_Final.tex, line 1421
[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.
Exact mathematical passage
F
fDefinition passage · 31 passages
Search aliases: f
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1195
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
f_{n,m}Definition passage · 3 passages
Search aliases: f_n,m
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1195
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
\frakaDefinition passage · 23 passages
Search aliases: fraka a mathfrak
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 729
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
\fraka\subseteq R_\infty
\frakpDefinition passage · 42 passages
Search aliases: frakp p mathfrak
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
HDefinition passage · 50 passages
Search aliases: H
Definition passage: Automorphic Hecke datum · Read passage · Paper_XII_Traditional_Final.tex, line 1294
[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at v is t…
Exact mathematical passage
H
\imDefinition passage · 4 passages
Search aliases: im operatorname
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
LDefinition passage · 15 passages
Search aliases: L
Definition passage: Algebraic Frobenius spine · Read passage · Paper_XII_Traditional_Final.tex, line 1421
[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.
Exact mathematical passage
L
\mathbf LDefinition passage · 10 passages
Search aliases: L mathbf
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\mathrm{loc}Definition passage · 12 passages
Search aliases: loc mathrm
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
\longleftarrowDefinition passage · 2 passages
Search aliases: longleftarrow
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
MDefinition passage · 119 passages
Search aliases: M
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 574
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
M
mDefinition passage · 82 passages
Search aliases: m
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1194
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
n\geq m
M_{\frakp}Definition passage · 10 passages
Search aliases: M_frakp M_ p frakp mathfrak
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
M_\inftyDefinition passage · 12 passages
Search aliases: M_∞ infty
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 729
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
M_\infty
\mathbinDefinition passage · 8 passages
Search aliases: mathbin
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\mathopDefinition passage · 8 passages
Search aliases: mathop
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
nDefinition passage · 67 passages
Search aliases: n
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1194
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
n\geq m
\notDefinition passage · 4 passages
Search aliases: not
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\OODefinition passage · 52 passages
Search aliases: OO O mathcal
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
PDefinition passage · 38 passages
Search aliases: P
Definition passage: Automorphic Hecke datum · Read passage · Paper_XII_Traditional_Final.tex, line 1296
efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial …
Exact mathematical passage
P_v(X)\in\TT[X].
P_vDefinition passage · 15 passages
Search aliases: P_v
Definition passage: Automorphic Hecke datum · Read passage · Paper_XII_Traditional_Final.tex, line 1296
efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial …
Exact mathematical passage
P_v(X)\in\TT[X].
RDefinition passage · 240 passages
Search aliases: R
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^\square
R^{\mathbf L}Definition passage · 2 passages
Search aliases: R^ L mathbf
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
R^{\mathrm{loc}}Definition passage · 11 passages
Search aliases: R^loc mathrm
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
R^{\mathrm{loc},\tau}Definition passage · 4 passages
Search aliases: R^loc,τ mathrm tau
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 219
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
R^\squareDefinition passage · 9 passages
Search aliases: R^square square
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 213
l deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR…
Exact mathematical passage
R^\square
R^{\tau}Definition passage · 3 passages
Search aliases: R^τ tau
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 219
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
R_v^\squareDefinition passage · 2 passages
Search aliases: R_v^square square
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^\square
R_v^{\tau_v}Definition passage · 2 passages
Search aliases: R_v^τ_v tau
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 212
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^{\tau_v}
R_\inftyDefinition passage · 22 passages
Search aliases: R_∞ infty
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 728
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
R_\infty
SDefinition passage · 57 passages
Search aliases: S
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
v\in S
\SpecDefinition passage · 27 passages
Search aliases: Spec operatorname
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\squareDefinition passage · 10 passages
Search aliases: square
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^\square
\SuppDefinition passage · 28 passages
Search aliases: Supp operatorname
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\Supp_RDefinition passage · 21 passages
Search aliases: Supp_R Supp operatorname
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\Supp_{R_\infty}Definition passage · 4 passages
Search aliases: Supp_R_∞ Supp infty operatorname infty
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1649
datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H supplied by the patched control isomorphism is the restriction…
Exact mathematical passage
\Supp_{R_\infty}(M_\infty)=\Spec R_\infty.
\TTDefinition passage · 60 passages
Search aliases: TT T mathbb
Definition passage: Automorphic Hecke datum · Read passage · Paper_XII_Traditional_Final.tex, line 1293
[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at v is t…
Exact mathematical passage
\TT
VDefinition passage · 78 passages
Search aliases: V
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
vDefinition passage · 51 passages
Search aliases: v
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
v\in S
V_1Definition passage · 1 passage
Search aliases: V_1
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
V_2Definition passage · 1 passage
Search aliases: V_2
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
V_3Definition passage · 1 passage
Search aliases: V_3
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
V_mDefinition passage · 8 passages
Search aliases: V_m
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1195
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
V_nDefinition passage · 1 passage
Search aliases: V_n
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1195
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
\wideotimesDefinition passage · 5 passages
Search aliases: wideotimes mathbinwidehat⊗ mathbin widehat otimes
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
\wideotimes_{R^{\mathrm{loc}}}Definition passage · 2 passages
Search aliases: wideotimes_R^loc mathbinwidehat⊗_R^loc wideotimes mathrm mathbin widehat otimes mathrm
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 219
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
\wideotimes_{v\in S,\OO}Definition passage · 2 passages
Search aliases: wideotimes_v∈ S,OO mathbinwidehat⊗_v∈ S, O wideotimes in OO mathbin widehat otimes in mathcal
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
XDefinition passage · 60 passages
Search aliases: X
Definition passage: Automorphic Hecke datum · Read passage · Paper_XII_Traditional_Final.tex, line 1296
efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial …
Exact mathematical passage
P_v(X)\in\TT[X].
xDefinition passage · 203 passages
Search aliases: x
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1640
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
x:R\to E'
X_{\lambda,1}Definition passage · 2 passages
Search aliases: X_λ,1 lambda
Definition passage: Finite companion forest · Read passage · Paper_XII_Traditional_Final.tex, line 1430
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
X_{\lambda,2}Definition passage · 1 passage
Search aliases: X_λ,2 lambda
Definition passage: Finite companion forest · Read passage · Paper_XII_Traditional_Final.tex, line 1430
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
X_{\lambda,3}Definition passage · 1 passage
Search aliases: X_λ,3 lambda
Definition passage: Finite companion forest · Read passage · Paper_XII_Traditional_Final.tex, line 1430
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
YDefinition passage · 27 passages
Search aliases: Y
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
Y_mDefinition passage · 8 passages
Search aliases: Y_m
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
\kappaDefinition passage · 9 passages
Search aliases: κ kappa
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\lambdaDefinition passage · 66 passages
Search aliases: λ lambda
Definition passage: Finite companion forest · Read passage · Paper_XII_Traditional_Final.tex, line 1429
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
\lambda
\lambda^mDefinition passage · 1 passage
Search aliases: λ^m lambda
Definition passage: Finite companion forest · Read passage · Paper_XII_Traditional_Final.tex, line 1438
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
\lambda^m
\rhoDefinition passage · 50 passages
Search aliases: ρ rho
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1642
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
\rho_R
\rho_RDefinition passage · 11 passages
Search aliases: ρ_R rho
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1642
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
\rho_R
\rho_xDefinition passage · 28 passages
Search aliases: ρ_x rho
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1657
phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and ρ_x=x∘ρ_R with x(P_v(X)) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplificat…
Exact mathematical passage
\rho_x=x\circ\rho_R
\tauDefinition passage · 11 passages
Search aliases: τ tau
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 212
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^{\tau_v}
\tau_vDefinition passage · 2 passages
Search aliases: τ_v tau
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 212
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^{\tau_v}
\toDefinition passage · 47 passages
Search aliases: → to
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1195
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
\inDefinition passage · 53 passages
Search aliases: ∈ in
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 211
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
v\in S
\notinDefinition passage · 2 passages
Search aliases: ∉ notin
Definition passage: Algebraic Frobenius spine · Read passage · Paper_XII_Traditional_Final.tex, line 1425
[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.
Exact mathematical passage
v\notin S
\circDefinition passage · 16 passages
Search aliases: ∘ circ
Definition passage: Ordinary completion datum at a point · Read passage · Paper_XII_Traditional_Final.tex, line 1657
phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and ρ_x=x∘ρ_R with x(P_v(X)) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplificat…
Exact mathematical passage
\rho_x=x\circ\rho_R
\inftyDefinition passage · 28 passages
Search aliases: ∞ infty
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 728
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
R_\infty
\simeqDefinition passage · 5 passages
Search aliases: ≃ simeq
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\congDefinition passage · 67 passages
Search aliases: ≅ cong
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 730
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
R\cong R_\infty/\fraka, \qquad M\cong M_\infty/\fraka M_\infty.
\neqDefinition passage · 18 passages
Search aliases: ≠ neq
Definition passage: Support and near faithfulness · Read passage · Paper_XII_Traditional_Final.tex, line 575
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\geqDefinition passage · 13 passages
Search aliases: ≥ geq
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1194
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
n\geq m
\subseteqDefinition passage · 17 passages
Search aliases: ⊆ subseteq
Definition passage: Ordinary patching datum · Read passage · Paper_XII_Traditional_Final.tex, line 729
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
\fraka\subseteq R_\infty
\otimesDefinition passage · 30 passages
Search aliases: ⊗ otimes
Definition passage: Derived support · Read passage · Paper_XII_Traditional_Final.tex, line 812
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\bigcapDefinition passage · 7 passages
Search aliases: ⋂ bigcap
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
\bigcap_{n\geq m}Definition passage · 1 passage
Search aliases: ⋂_n≥ m bigcap geq
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
\cdotsDefinition passage · 11 passages
Search aliases: ⋯ cdots
Definition passage: Stable core · Read passage · Paper_XII_Traditional_Final.tex, line 1191
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
\longrightarrowDefinition passage · 14 passages
Search aliases: ⟶ longrightarrow
Definition passage: Global deformation datum · Read passage · Paper_XII_Traditional_Final.tex, line 214
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
*Introduction passage · 7 passages
Search aliases: *
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 1738
eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ T satisfying (f^*)^-1(Supp_R(M))⊆Supp_ T(H). Use ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusio…
Exact mathematical passage
(f^*)^{-1}\bigl(\Supp_R(M)\bigr)\subseteq\Supp_{\TT}(H).
+Introduction passage · 71 passages
Search aliases: +
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
Exact mathematical passage
\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
aIntroduction passage · 56 passages
Search aliases: a
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 524
For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the commutative group algebra.
Exact mathematical passage
ma
f^*Introduction passage · 2 passages
Search aliases: f^*
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 1738
eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ T satisfying (f^*)^-1(Supp_R(M))⊆Supp_ T(H). Use ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusio…
Exact mathematical passage
(f^*)^{-1}\bigl(\Supp_R(M)\bigr)\subseteq\Supp_{\TT}(H).
\FFIntroduction passage · 29 passages
Search aliases: FF F mathbb
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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\FF_p
\FF_pIntroduction passage · 2 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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\FF_p
\frakmIntroduction passage · 24 passages
Search aliases: frakm m mathfrak
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\frakm_{\OO}Introduction passage · 1 passage
Search aliases: frakm_OO m_ O frakm OO mathfrak mathcal
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\frakm_{R_i}Introduction passage · 1 passage
Search aliases: frakm_R_i m_R_i frakm mathfrak
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
GIntroduction passage · 15 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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e\in k[G]
gIntroduction passage · 28 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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ge
IIntroduction passage · 49 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
iIntroduction passage · 52 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 272
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
I_iIntroduction passage · 11 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
kIntroduction passage · 57 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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e\in k[G]
pIntroduction passage · 25 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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p
R_iIntroduction passage · 9 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 272
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
sIntroduction passage · 21 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 272
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
\Supp_{\TT}Introduction passage · 3 passages
Search aliases: Supp_TT Supp_ T Supp TT operatorname mathbb
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 1738
eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ T satisfying (f^*)^-1(Supp_R(M))⊆Supp_ T(H). Use ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusio…
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(f^*)^{-1}\bigl(\Supp_R(M)\bigr)\subseteq\Supp_{\TT}(H).
WIntroduction passage · 30 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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W
\epsilonIntroduction passage · 15 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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\epsilon:k[G]\to k
\LambdaIntroduction passage · 25 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 524
For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the commutative group algebra.
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M\otimes_{\Lambda,\epsilon}\OO
\sumIntroduction passage · 9 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\sum_iIntroduction passage · 2 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\neIntroduction passage · 13 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 519
The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit counting formula implies that its fixed point set also has cardinality divisible by p. Since zero is fixed, there is a nonzero fixed vector v∈ W.
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e\ne0
\bigotimesIntroduction passage · 3 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 272
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
\bigotimes_{i=1}^{s}Introduction passage · 2 passages
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Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 272
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
\otimes_{\Lambda,\epsilon}Introduction passage · 6 passages
Search aliases: ⊗_Λ,ϵ otimes Lambda epsilon
Introduction passage · Read passage · Paper_XII_Traditional_Final.tex, line 524
For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the commutative group algebra.
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M\otimes_{\Lambda,\epsilon}\OO
AUse in source · 38 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 297
Let f:R^loc→ A be the point. Since A is Hausdorff and f is continuous, ker f is closed. Thus it contains the closed ideal J if and only if it contains the algebraic ideal generated by the images of the I_i. This is equivalent to every local restriction killing I_i. The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If f reflects units, lift an element of the quotient to the or…
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f:R^{\mathrm{loc}}\to A
\aUse in source · 1 passage
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Use in source · derived_fibers.tex, line 7
node[anchor=east] at (-.3,2.25) point a; node[anchor=east] at (-.3,1.4) dim H^0(C_a); node[anchor=east] at (-.3,.55) dim H^1(C_a); foreach x/a in 0/-2,1.5/-1,3/0,4.5/1,6/2 node at (x+.4,2.25) a; draw[black!35] (x,1.05) rectangle ++(.8,.7);draw[black!35] (x,.2) rectangle ++(.8,.7); foreach x in 0,1.5,4.5,6node[text=black!45] at (x+.4,1.4) 0;node[text=black!45] at (x+.4,.55) 0; fill[black!12] (3,1.05) rectangle ++(.8,.7);fill[black!12] (3,.2) rectangle ++(.8,.7); node at (3.4,1.4) 1;node at (3.4,.…
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\a
\mathfrak aUse in source · 3 passages
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Use in source · pages.tex, line 649
-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition: S=⊗_i=1^s R_i, a=m_ OS+∑_im_R_iS, L=∑_i I_iS. The comparison uses the a-adic additive uniformity on S, the image ideal topology on S/L…
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S=\bigotimes_{i=1}^s R_i,\qquad \mathfrak a=\m_{\OO}S+\sum_i\m_{R_i}S,\qquad L=\sum_i I_iS.
A^2Use in source · 1 passage
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Use in source · pages.tex, line 773
% --------------------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22),
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A^2-I
a^2Use in source · 3 passages
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Use in source · pages.tex, line 107
u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Longleftrightarrow a=c=0, d=b.
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a^2=0,\quad c=a,\quad d=b \quad\Longleftrightarrow\quad a=c=0,\ d=b.
a_{00}Use in source · 1 passage
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Use in source · pages.tex, line 82
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
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f\longmapsto a_{00}+a_{10}\varepsilon.
a_{10}Use in source · 1 passage
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Use in source · pages.tex, line 82
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
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f\longmapsto a_{00}+a_{10}\varepsilon.
a_{11}Use in source · 2 passages
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Use in source · pages.tex, line 774
-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22),
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B=\ZZ_5[a_{11},a_{12},a_{21},a_{22}],
a_{11}^2Use in source · 1 passage
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Use in source · pages.tex, line 778
Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22), a_21(a_11+a_22), · a_21a_12+a_22^2-1. For D and every conjugate QDQ^-1 these equations vanish. The conditi…
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\begin{aligned} a_{11}^2+a_{12}a_{21}-1,&\qquad a_{12}(a_{11}+a_{22}),\\ a_{21}(a_{11}+a_{22}),&\qquad a_{21}a_{12}+a_{22}^2-1. \end{aligned}
a_{12}Use in source · 3 passages
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Use in source · pages.tex, line 774
-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22),
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B=\ZZ_5[a_{11},a_{12},a_{21},a_{22}],
a_{21}Use in source · 2 passages
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Use in source · pages.tex, line 774
-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22),
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B=\ZZ_5[a_{11},a_{12},a_{21},a_{22}],
a_{22}Use in source · 2 passages
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Use in source · pages.tex, line 774
-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22),
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B=\ZZ_5[a_{11},a_{12},a_{21},a_{22}],
a_{22}^2Use in source · 1 passage
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Use in source · pages.tex, line 778
Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generated by a_11^2+a_12a_21-1, · a_12(a_11+a_22), a_21(a_11+a_22), · a_21a_12+a_22^2-1. For D and every conjugate QDQ^-1 these equations vanish. The conditi…
Exact mathematical passage
\begin{aligned} a_{11}^2+a_{12}a_{21}-1,&\qquad a_{12}(a_{11}+a_{22}),\\ a_{21}(a_{11}+a_{22}),&\qquad a_{21}a_{12}+a_{22}^2-1. \end{aligned}
A_{\frakp}Use in source · 1 passage
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 623
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
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A_{\frakp}
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 465
Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.
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\epsilon:\Lambda\longrightarrow\OO, \qquad \sum_{g\in\Delta}a_gg\longmapsto\sum_{g\in\Delta}a_g.
a_{ij}Use in source · 2 passages
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Use in source · pages.tex, line 81
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
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f=\sum_{i,j\ge0}a_{ij}x^iy^j
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1214
Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.
Exact mathematical passage
A_n = \left\{ z\in\im(f_{n,m+1}): f_{m+1,m}(z)=y \right\}.
\absUse in source · 1 passage
Search aliases: abs
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 511
In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique maximal ideal.
Exact mathematical passage
\prod_ip^{n_i}=\abs{\Delta}
\adUse in source · 6 passages
Search aliases: ad operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 350
[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.
Exact mathematical passage
H^1(G,\ad^0V)
\operatorname{ad}Use in source · 1 passage
Search aliases: ad operatorname
Use in source · pages.tex, line 693
headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.
Exact mathematical passage
H^0(S_3,\operatorname{ad}^0V)=0
\ad^0Use in source · 6 passages
Search aliases: ad^0 ad operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 350
[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.
Exact mathematical passage
H^1(G,\ad^0V)
\operatorname{ad}^0Use in source · 1 passage
Search aliases: ad^0 operatorname
Use in source · pages.tex, line 693
headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.
Exact mathematical passage
H^0(S_3,\operatorname{ad}^0V)=0
\arraybackslashUse in source · 1 passage
Search aliases: arraybackslash
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
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3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
BUse in source · 29 passages
Search aliases: B
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 311
Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/o…
Exact mathematical passage
B=R^\square
bUse in source · 31 passages
Search aliases: b
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 394
The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles ar…
Exact mathematical passage
(a,b)\longmapsto(a,-a+b,-b)
b^2Use in source · 1 passage
Search aliases: b^2
Use in source · pages.tex, line 286
headAn isomorphism changes coordinates, not occupancy Over a field of characteristic other than two, the isomorphism k[x,y]xrightarrow∼k[a,b], x↦ a+b, y↦ a-b sends the Fitting ideals to (a^2-b^2) and (a,b). The two axes become the two lines a=b and a=-b. No previously missing module vector is created by this relabeling.
Exact mathematical passage
(a^2-b^2)
B_nUse in source · 1 passage
Search aliases: B_n
Use in source · pages.tex, line 131
% --------------------------------------------------------------------------- vpage5Completion retains every precisionMODEL A / COMPATIBLE COEFFICIENTSMain text: Theorem 3.1; reconstruction and finite occupancy in Section 10 p12-route:visual-5Main statement 10.5 For B= Z_5[ε]/(ε^2), consider the finite quotient rings B_n=( Z/5^n Z)[ε]/(ε^2), |B_n|=5^2n, B≅varprojlim_n B_n. A point of the limit is a single compatible choice at every level, not unrelated choices from the nonempty finite rings.
Exact mathematical passage
B_n=(\ZZ/5^n\ZZ)[\varepsilon]/(\varepsilon^2), \qquad |B_n|=5^{2n},\qquad B\cong\varprojlim_n B_n.
B_\lambdaUse in source · 4 passages
Search aliases: B_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1485
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
Exact mathematical passage
B_\lambda=\OO_\lambda[x_{g,a,b}:g\in\Gamma,\ a,b\in D].
\bar gUse in source · 1 passage
Search aliases: bar g bar
Use in source · pages.tex, line 715
headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.
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g\mapsto\bar g
\bar XUse in source · 2 passages
Search aliases: bar X bar
Use in source · pages.tex, line 715
headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.
Exact mathematical passage
\bar X
\bar X^2Use in source · 1 passage
Search aliases: bar X^2 bar
Use in source · pages.tex, line 715
headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.
Exact mathematical passage
\bar X^2=0
\bmodUse in source · 4 passages
Search aliases: bmod
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1482
us elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which all these finite representations are trivial. This is an inertia condition when H_λ is the specified inertia subgroup.
Exact mathematical passage
P_{\lambda,v}\bmod I_\lambda^n
\bottomruleUse in source · 1 passage
Search aliases: bottomrule
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
cUse in source · 14 passages
Search aliases: c
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 410
Write W(a,b,c)=a · b c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.
Exact mathematical passage
W(a,b,c)=\begin{pmatrix}a&b\\c&-a\end{pmatrix}, \qquad Z=W(2,3,4),\quad A=W(1,1,2),\quad B=W(1,5,6).
C_2Use in source · 12 passages
Search aliases: C_2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1579
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
C_2=\langle g\rangle
C_4Use in source · 2 passages
Search aliases: C_4
Use in source · pages.tex, line 702
% --------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-1. Then F_2[Δ]≅ F_2[X,Y]/(X^4,Y^2). The identity (1+X)^4-1=X^4 and its Y analogue provide the relations. The eight surviving monomials provide the exa…
Exact mathematical passage
\Delta=C_4\times C_2
C_aUse in source · 3 passages
Search aliases: C_a
Use in source · derived_fibers.tex, line 4
node[anchor=east] at (-.3,2.25) point a; node[anchor=east] at (-.3,1.4) dim H^0(C_a); node[anchor=east] at (-.3,.55) dim H^1(C_a); foreach x/a in 0/-2,1.5/-1,3/0,4.5/1,6/2 node at (x+.4,2.25) a; draw[black!35] (x,1.05) rectangle ++(.8,.7);draw[black!35] (x,.2) rectangle ++(.8,.7); foreach x in 0,1.5,4.5,6node[text=black!45] at (x+.4,1.4) 0;node[text=black!45] at (x+.4,.55) 0; fill[black!12] (3,1.05) rectangle ++(.8,.7);fill[black!12] (3,.2) rectangle ++(.8,.7); node at (3.4,1.4) 1;node at (3.4,.…
Exact mathematical passage
\dim H^0(C_a)
C_{p^{n_i}}Use in source · 1 passage
Search aliases: C_p^n_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
\cokerUse in source · 3 passages
Search aliases: coker operatorname
Use in source · pages.tex, line 212
% --------------------------------------------------------------------------- vpage7The same support can hide more algebraMODEL B / NILPOTENT STRUCTUREMain text: Theorems 6.3--6.4; distinguish an ideal from its vanishing set p12-route:visual-7Main statement 6.4 Compare the previous module with a slightly different presentation: M=cokerx · 0 0 · y, M'=cokerx^2 · 0 0 · y. Their support sets coincide. Their residue-field fiber dimensions also coincide at every prime. But their Fitting ideals do not…
Exact mathematical passage
M=\coker\begin{pmatrix}x&0\\0&y\end{pmatrix}, \qquad M'=\coker\begin{pmatrix}x^2&0\\0&y\end{pmatrix}.
DUse in source · 25 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 432
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
D
dUse in source · 22 passages
Search aliases: d
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 527
[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.
Exact mathematical passage
d
D_2Use in source · 2 passages
Search aliases: D_2
Use in source · pages.tex, line 124
headAn obstruction you can calculate There are 5^4=625 candidate D_3-points and exactly five accepted points. Over D_2= F_5[δ]/(δ^2), however, the five maps u=v=aδ all satisfy the equations. Reduction of every accepted D_3-point gives the zero D_2-point. The four nonzero tangent directions therefore do not lift to order three.
Exact mathematical passage
D_2=\FF_5[\delta]/(\delta^2)
D_3Use in source · 3 passages
Search aliases: D_3
Use in source · pages.tex, line 100
ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Lon…
Exact mathematical passage
D_3=\FF_5[\delta]/(\delta^3)
\detUse in source · 3 passages
Search aliases: det
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1355
g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\det\left(X-\rho_x(\Frob_v)\right)=x(P_v(X))
\operatorname{diag}Use in source · 5 passages
Search aliases: diag operatorname
Use in source · pages.tex, line 260
% --------------------------------------------------------------------------- vpage8Base change is an algebraic pullbackMODEL B / SPECIALIZATION AND TRANSPORTMain text: Proposition 6.5 and Theorem 7.1 p12-route:visual-8Main statement 6.5 Return to M=cokerdiag(x,y) over R=k[x,y]. Pull back along the diagonal R→ k[t], x↦ t, y↦ t. The base-changed presentation is literal: M⊗_R k[t]≅cokert · 0
Exact mathematical passage
M=\coker\operatorname{diag}(x,y)
\dimUse in source · 6 passages
Search aliases: dim
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 628
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\dim_{E'}Use in source · 1 passage
Search aliases: dim_E' dim
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1092
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\dim_{\kappa(\frakp)}Use in source · 1 passage
Search aliases: dim_κ(frakp) dim_κ( p) dim kappa frakp dim kappa mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 628
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
E'^rUse in source · 5 passages
Search aliases: E'^r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1555
Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.
Exact mathematical passage
W = \operatorname{span} \left\{ (t_1(\omega),\ldots,t_r(\omega)):\omega\in\Omega \right\} \subseteq E'^r.
e^{-1}Use in source · 2 passages
Search aliases: e^-1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1388
[Derived pointwise realization] Let e:R≅ T, and let C be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for C, supported in degrees [a,b]. For every characteristic zero point x:R→ E', write y=x∘ e^-1. Then some integer i∈[a,b] has H^i(Cmathbinmathop⊗limits_ T,y^ LE')≠0. Every t∈ T acts on this cohomology through y(t).
Exact mathematical passage
y=x\circ e^{-1}
e^2Use in source · 3 passages
Search aliases: e^2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 331
Let e^2=e. Since e+(1-e)=1, at least one of e and 1-e is a unit. If e is a unit, then e=1. If 1-e is a unit, then e=0.
Exact mathematical passage
e^2=e
e_{\mathfrak n}Use in source · 1 passage
Search aliases: e_ n mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1032
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
e_+Use in source · 8 passages
Search aliases: e_+
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1580
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ e_+=\frac{1+g}{2}, \qquad e_-=\frac{1-g}{2}. }
e_-Use in source · 6 passages
Search aliases: e_-
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1580
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ e_+=\frac{1+g}{2}, \qquad e_-=\frac{1-g}{2}. }
e_1Use in source · 9 passages
Search aliases: e_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1149
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
e_1,e_2,\ldots
e_2Use in source · 7 passages
Search aliases: e_2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1149
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
e_1,e_2,\ldots
e_3Use in source · 4 passages
Search aliases: e_3
Use in source · core_nilpotent.tex, line 39
times; draw[black!35,fill=white] (0.66,0.0) rectangle ++(.55,.55); node[text=black!30] at (0.935,0.275) ×; draw[black!35,fill=white] (1.32,0.0) rectangle ++(.55,.55); node[text=black!30] at (1.5950000000000002,0.275) ×; node[font=footnotesize] at (0.275,3.3) e_1; node[font=footnotesize] at (0.935,3.3) e_2; node[font=footnotesize] at (1.5950000000000002,3.3) e_3;
Exact mathematical passage
e_3
e_4Use in source · 5 passages
Search aliases: e_4
Use in source · finite_forest.tex, line 6
node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: …
Exact mathematical passage
e_4
e_5Use in source · 4 passages
Search aliases: e_5
Use in source · finite_forest.tex, line 6
node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: …
Exact mathematical passage
e_5,e_6,\ldots
e_6Use in source · 2 passages
Search aliases: e_6
Use in source · finite_forest.tex, line 6
node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: …
Exact mathematical passage
e_5,e_6,\ldots
e_7Use in source · 1 passage
Search aliases: e_7
Use in source · infinite_tail.tex, line 101
t (3.86,0.2) ⋯; node[font=scriptsize] at (0.195,3.34) e_1; node[font=scriptsize] at (0.665,3.34) e_2; node[font=scriptsize] at (1.135,3.34) e_3; node[font=scriptsize] at (1.605,3.34) e_4; node[font=scriptsize] at (2.0749999999999997,3.34) e_5; node[font=scriptsize] at (2.5449999999999995,3.34) e_6; node[font=scriptsize] at (3.0149999999999997,3.34) e_7; node[font=scriptsize] at (3.485,3.34) e_8;
Exact mathematical passage
e_7
e_8Use in source · 1 passage
Search aliases: e_8
Use in source · infinite_tail.tex, line 102
at (0.195,3.34) e_1; node[font=scriptsize] at (0.665,3.34) e_2; node[font=scriptsize] at (1.135,3.34) e_3; node[font=scriptsize] at (1.605,3.34) e_4; node[font=scriptsize] at (2.0749999999999997,3.34) e_5; node[font=scriptsize] at (2.5449999999999995,3.34) e_6; node[font=scriptsize] at (3.0149999999999997,3.34) e_7; node[font=scriptsize] at (3.485,3.34) e_8;
Exact mathematical passage
e_8
e_{\frakm}Use in source · 10 passages
Search aliases: e_frakm e_ m frakm mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1031
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
(e_{\frakm})
e_jUse in source · 1 passage
Search aliases: e_j
Use in source · pages.tex, line 568
captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_tail captiontextRow m retains all basis vectors e_j with j≥ m. Every row continues infinitely to the right. The display is a finite window, not a truncation of the actual carrier.
Exact mathematical passage
e_j
e_mUse in source · 9 passages
Search aliases: e_m
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1150
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
e_{m+1}e_m=e_me_{m+1}=e_{m+1}
e_{\m}Use in source · 1 passage
Search aliases: e_m m
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
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3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
e_{m+1}Use in source · 5 passages
Search aliases: e_m+1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1150
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
e_{m+1}e_m=e_me_{m+1}=e_{m+1}
e_{m+2}Use in source · 1 passage
Search aliases: e_m+2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1256
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
e_m,e_{m+1},e_{m+2},\ldots.
e_nUse in source · 2 passages
Search aliases: e_n
Use in source · pages.tex, line 730
------------------------------------- vpage22A finite forest selects a branchFINITE OCCUPANCY / MAPS NEED NOT BE SURJECTIVEMain text: Theorem 10.3 and Corollary 10.4 p12-route:visual-22Main statement 10.3 A finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system: X_1={r}, X_2={a,b}, X_3={c,d}, X_n={e_n} (n≥4). Both a,b reduce to r; both c,d reduce to a; e_4 reduces to c; and e_n+1 reduc…
Exact mathematical passage
X_1=\{r\},\quad X_2=\{a,b\},\quad X_3=\{c,d\},\quad X_n=\{e_n\}\quad(n\ge4).
e_{n+1}Use in source · 1 passage
Search aliases: e_n+1
Use in source · pages.tex, line 734
orollary 10.4 p12-route:visual-22Main statement 10.3 A finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system: X_1={r}, X_2={a,b}, X_3={c,d}, X_n={e_n} (n≥4). Both a,b reduce to r; both c,d reduce to a; e_4 reduces to c; and e_n+1 reduces to e_n for every n≥4.
Exact mathematical passage
e_{n+1}
e_\lambdaUse in source · 3 passages
Search aliases: e_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
i_\lambda:\OO_\lambda\longrightarrow K_\lambda, \qquad e_\lambda:L\longrightarrow K_\lambda
f^{-1}Use in source · 1 passage
Search aliases: f^-1
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
F^hUse in source · 1 passage
Search aliases: F^h
Use in source · pages.tex, line 521
-------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions visible at level m after looking h levels ahead form im(F^h).
Exact mathematical passage
\im(F^h)
f_{m+1,m}Use in source · 1 passage
Search aliases: f_m+1,m
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1214
Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.
Exact mathematical passage
A_n = \left\{ z\in\im(f_{n,m+1}): f_{m+1,m}(z)=y \right\}.
f_{n,m+1}Use in source · 1 passage
Search aliases: f_n,m+1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1214
Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.
Exact mathematical passage
A_n = \left\{ z\in\im(f_{n,m+1}): f_{m+1,m}(z)=y \right\}.
\FF_2Use in source · 8 passages
Search aliases: FF_2 F_2 FF mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1588
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ \FF_2[C_2] \cong \FF_2[X]/(X^2), \qquad X=g-1. }
\FF_2^2Use in source · 1 passage
Search aliases: FF_2^2 F_2^2 FF mathbb
Use in source · pages.tex, line 506
headA trace can vanish while a packet is occupied The identity projector on F_2^2 has rank two and trace 2=0. Hence nonzero rank and nonzero trace are not equivalent in this characteristic. The localization and rank criteria still detect the nonzero space. The characteristic-zero qualification in Corollary 8.4 is essential.
Exact mathematical passage
\FF_2^2
\FF_5Use in source · 5 passages
Search aliases: FF_5 F_5 FF mathbb
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\FF_7Use in source · 12 passages
Search aliases: FF_7 F_7 FF mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 355
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
k=\FF_7
\FF_7^3Use in source · 1 passage
Search aliases: FF_7^3 F_7^3 FF mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 397
The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles ar…
Exact mathematical passage
\FF_7^3
\FittUse in source · 9 passages
Search aliases: Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 608
[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).
Exact mathematical passage
\boxed{ \Supp_R(M) = V\bigl(\Fitt_0^R(M)\bigr). }
\Fitt_0Use in source · 2 passages
Search aliases: Fitt_0 Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 618
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
\Fitt_0(0)=R
\Fitt_0^RUse in source · 2 passages
Search aliases: Fitt_0^R Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 608
[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).
Exact mathematical passage
\boxed{ \Supp_R(M) = V\bigl(\Fitt_0^R(M)\bigr). }
\Fitt_1Use in source · 1 passage
Search aliases: Fitt_1 Fitt operatorname
Use in source · pages.tex, line 181
[c].48linewidth vfig[.98]support_axes hfill [c].48linewidth sffamilybfseries The minor calculation Fitt_0(M) · =(xy), Fitt_1(M) · =(x,y), Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full algebraic loc…
Exact mathematical passage
\begin{aligned} \Fitt_0(M)&=(xy),\\ \Fitt_1(M)&=(x,y),\\ \Fitt_j(M)&=R\quad(j\ge2). \end{aligned}
\Fitt_jUse in source · 1 passage
Search aliases: Fitt_j Fitt operatorname
Use in source · pages.tex, line 181
[c].48linewidth vfig[.98]support_axes hfill [c].48linewidth sffamilybfseries The minor calculation Fitt_0(M) · =(xy), Fitt_1(M) · =(x,y), Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full algebraic loc…
Exact mathematical passage
\begin{aligned} \Fitt_0(M)&=(xy),\\ \Fitt_1(M)&=(x,y),\\ \Fitt_j(M)&=R\quad(j\ge2). \end{aligned}
\Fitt_j^RUse in source · 2 passages
Search aliases: Fitt_j^R Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 657
[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.
Exact mathematical passage
\boxed{ \Fitt_j^S(M\otimes_RS) = \Fitt_j^R(M)S. }
\Fitt_j^SUse in source · 1 passage
Search aliases: Fitt_j^S Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 657
[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.
Exact mathematical passage
\boxed{ \Fitt_j^S(M\otimes_RS) = \Fitt_j^R(M)S. }
\Fitt_j^{\TT}Use in source · 1 passage
Search aliases: Fitt_j^TT Fitt_j^ T Fitt TT operatorname mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 934
[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).
Exact mathematical passage
\boxed{ \varphi \left( \Fitt_j^R(M) \right) = \Fitt_j^{\TT}(M). }
\Fitt_{r-1}Use in source · 1 passage
Search aliases: Fitt_r-1 Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 648
Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equivalent to the vanishing of every minor of size n-r+1. Those minors generate Fitt_r-1^R(M).
Exact mathematical passage
\Fitt_{r-1}(M)=R
\Fitt_{r-1}^RUse in source · 2 passages
Search aliases: Fitt_r-1^R Fitt operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 635
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\boxed{ \left\{ \frakp\in\Spec R: \mu_{\frakp}(M)\geq r \right\} = V\bigl(\Fitt_{r-1}^R(M)\bigr). }
\frakm_AUse in source · 1 passage
Search aliases: frakm_A m_A frakm mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 315
and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The constructi…
Exact mathematical passage
\frakm_A
\frakm_BUse in source · 2 passages
Search aliases: frakm_B m_B frakm mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 315
Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/o…
Exact mathematical passage
\frakm_B
\FrobUse in source · 1 passage
Search aliases: Frob operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1355
g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\det\left(X-\rho_x(\Frob_v)\right)=x(P_v(X))
\Frob_vUse in source · 1 passage
Search aliases: Frob_v Frob operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1355
g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\det\left(X-\rho_x(\Frob_v)\right)=x(P_v(X))
G^2Use in source · 1 passage
Search aliases: G^2
Use in source · pages.tex, line 498
bottomrule hfill [c].47linewidth In the basis 1,δ, multiplication by g=1+δ is G=1 · 0 1 · 1, G^2=I over F_2. It is not the identity: G-I is nonzero and square-zero.
Exact mathematical passage
G=\begin{pmatrix}1&0\\1&1\end{pmatrix},\qquad G^2=I\quad\text{over }\FF_2.
g^2Use in source · 3 passages
Search aliases: g^2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1602
The displayed rational elements are the unique primitive idempotents because they project to the two characters of C_2. Their coefficients contain 1/2, so they are not integral. In characteristic two, (g-1)^2=g^2-2g+1=0. The map X↦ g-1 gives the displayed isomorphism. The quotient by (X) is F_2, and (X) is nilpotent. Thus the ring is local.
Exact mathematical passage
(g-1)^2=g^2-2g+1=0.
G_{F,S}Use in source · 5 passages
Search aliases: G_F,S
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 202
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow\GL_n(k)
\GLUse in source · 10 passages
Search aliases: GL operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 202
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow\GL_n(k)
\GL_2Use in source · 2 passages
Search aliases: GL_2 GL operatorname
Use in source · pages.tex, line 751
age23Classes are not matrix framesCOMPANION COMPLETION / ACTUAL REPRESENTATIVESMain text: revised Theorem 10.5 and its preceding admissibility hypotheses p12-route:visual-23Main statement 10.5 Let Γ=C_2 with the discrete topology, and represent its generator by D=diag(1,-1) over Z_5. At precision n, its conjugacy class is an admissible class in GL_2( Z/5^n Z). These classes are compatible. Arbitrary representatives need not be.
Exact mathematical passage
\GL_2(\ZZ/5^n\ZZ)
\GL_DUse in source · 3 passages
Search aliases: GL_D GL operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1473
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
\Gamma\longrightarrow\GL_D(\OO_\lambda/I_\lambda^n).
\GL_nUse in source · 4 passages
Search aliases: GL_n GL operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 202
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow\GL_n(k)
hUse in source · 23 passages
Search aliases: h
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 404
A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomology vanishings.
Exact mathematical passage
u=-6^{-1}\sum_{h\in S_3}f(h).
H^0Use in source · 6 passages
Search aliases: H^0
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 378
[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.
Exact mathematical passage
H^0(S_3,\ad^0V)=0, \qquad H^1(S_3,\FF_7)=0, \qquad H^1(S_3,\ad^0V)=0,
H^1Use in source · 9 passages
Search aliases: H^1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 350
[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.
Exact mathematical passage
H^1(G,k)
H^iUse in source · 3 passages
Search aliases: H^i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 825
[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].
Exact mathematical passage
\Supp_R(C) = \bigcup_i\Supp_R H^i(C).
H_xUse in source · 3 passages
Search aliases: H_x
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1323
[Support gives a Hecke eigensystem] Let R≅ T, let H be a finite T module, and let x:R→ E' be a field valued point. If ker x∈Supp_R(H), then H_x:=H⊗_ T,xE'≠0. Every element of T acts on H_x through the scalar prescribed by x.
Exact mathematical passage
\boxed{ H_x:=H\otimes_{\TT,x}E'\neq0. }
H_\lambdaUse in source · 4 passages
Search aliases: H_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1482
y an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which all these finite representations are trivial. This is an inertia condition when H_λ is the specified inertia subgroup.
Exact mathematical passage
H_\lambda\leq\Gamma
\headUse in source · 1 passage
Search aliases: head
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\hlineUse in source · 1 passage
Search aliases: hline
Use in source · pages.tex, line 401
in statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ underbrace Q[ε]/(ε^2)_T_+, z=1+ε × underbrace Q[η]/(η^3)_T_-, z=-1+η. Multiplication by z is an operator H. In the ordered local bases 1,ε and 1,η,η^2, its matrix is H=(cc|ccc 1 · 0 · 0 · 0 · 0 1 · 1 · 0 · 0 · 0 0 · 0 · -1 · 0 · 0 0 · 0 · 1 · -1 · 0 0 · 0 · 0 · 1 · -1 ).
Exact mathematical passage
H=\left(\begin{array}{cc|ccc} 1&0&0&0&0\\1&1&0&0&0\\\hline 0&0&-1&0&0\\0&0&1&-1&0\\0&0&0&1&-1 \end{array}\right).
I_1Use in source · 3 passages
Search aliases: I_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 262
gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
\boxed{ (R_1/I_1)\wideotimes_{\OO}\cdots\wideotimes_{\OO}(R_s/I_s) \cong R^{\mathrm{loc}}/J. }
I_2Use in source · 2 passages
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Use in source · pages.tex, line 66
% --------------------------------------------------------------------------- vpage3Local conditions remove directionsMODEL A / A FORMAL THICKENINGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_1/I_1…
Exact mathematical passage
R_1=\OO[[x]],\quad I_1=(x^2),\qquad R_2=\OO[[y]],\quad I_2=(y).
I_{\frakp}Use in source · 3 passages
Search aliases: I_frakp I_ p frakp mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 717
Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.
Exact mathematical passage
(M/IM)_{\frakp} \cong M_{\frakp}/I_{\frakp}M_{\frakp}.
I_sUse in source · 1 passage
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 262
gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
\boxed{ (R_1/I_1)\wideotimes_{\OO}\cdots\wideotimes_{\OO}(R_s/I_s) \cong R^{\mathrm{loc}}/J. }
I_\lambdaUse in source · 4 passages
Search aliases: I_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1465
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
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I_\lambda
i_\lambdaUse in source · 5 passages
Search aliases: i_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
i_\lambda:\OO_\lambda\longrightarrow K_\lambda, \qquad e_\lambda:L\longrightarrow K_\lambda
I_\lambda^nUse in source · 3 passages
Search aliases: I_λ^n lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1473
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
\Gamma\longrightarrow\GL_D(\OO_\lambda/I_\lambda^n).
JUse in source · 29 passages
Search aliases: J
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 261
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
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J
jUse in source · 20 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 656
[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.
Exact mathematical passage
j\geq0
j^2Use in source · 1 passage
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Use in source · pages.tex, line 139
headTwo explicit 5-adic numbers Choose the canonical digits α_j=(j^2+1)bmod5, β_j=(2j+1)bmod5, with values in {0,1,2,3,4}. Set a=∑_j≥0α_j5^j and b=∑_j≥0β_j5^j. The element a+ε b is determined by the following coefficient array.
Exact mathematical passage
\alpha_j=(j^2+1)\bmod5,\qquad \beta_j=(2j+1)\bmod5,
J_{\lambda,u}Use in source · 5 passages
Search aliases: J_λ,u lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1488
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
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J_{\lambda,u}\subseteq B_\lambda
KUse in source · 4 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
i_\lambda:\OO_\lambda\longrightarrow K_\lambda, \qquad e_\lambda:L\longrightarrow K_\lambda
k^3Use in source · 2 passages
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Use in source · pages.tex, line 521
% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions visible at level m after looking h levels ahead form im(F^h).
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V_m=k^3
k^5Use in source · 1 passage
Search aliases: k^5
Use in source · pages.tex, line 561
[t].44linewidthcentering sffamilybfseries Fixed carrier k^5 vfig[.75]nested_finite captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_tail captiontextRow m retains all basis vectors e_j with j≥ m. Every row continues infinitely to the right. The display is a finite window, not a trunca…
Exact mathematical passage
k^5
K_\lambdaUse in source · 4 passages
Search aliases: K_λ lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
i_\lambda:\OO_\lambda\longrightarrow K_\lambda, \qquad e_\lambda:L\longrightarrow K_\lambda
\kerUse in source · 10 passages
Search aliases: ker
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 297
Let f:R^loc→ A be the point. Since A is Hausdorff and f is continuous, ker f is closed. Thus it contains the closed ideal J if and only if it contains the algebraic ideal generated by the images of the I_i. This is equivalent to every local restriction killing I_i. The universal property of the actual ideal quotient gives the factors, and the quotient topology gives their continuity. Surjectivity of the quotient map gives uniqueness. If f reflects units, lift an element of the quotient to the or…
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\ker f
lUse in source · 1 passage
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Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
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3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\linewidthUse in source · 1 passage
Search aliases: linewidth
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\LongleftrightarrowUse in source · 3 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 977
[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).
Exact mathematical passage
\boxed{ M\otimes_{R,x}E'\neq0 \quad\Longleftrightarrow\quad \frakp\in\Supp_R(M). }
\LongrightarrowUse in source · 1 passage
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Use in source · pages.tex, line 814
1 · -1, det A=-2. It is invertible over Q. If the measured trace vector is (5,1), solve Am_1 m_2=5 1 Longrightarrow m_1 m_2=3 2.
Exact mathematical passage
A\begin{pmatrix}m_1\\m_2\end{pmatrix}=\begin{pmatrix}5\\1\end{pmatrix} \quad\Longrightarrow\quad \begin{pmatrix}m_1\\m_2\end{pmatrix}=\begin{pmatrix}3\\2\end{pmatrix}.
\mUse in source · 3 passages
Search aliases: m
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
M'Use in source · 4 passages
Search aliases: M'
Use in source · pages.tex, line 212
% --------------------------------------------------------------------------- vpage7The same support can hide more algebraMODEL B / NILPOTENT STRUCTUREMain text: Theorems 6.3--6.4; distinguish an ideal from its vanishing set p12-route:visual-7Main statement 6.4 Compare the previous module with a slightly different presentation: M=cokerx · 0 0 · y, M'=cokerx^2 · 0 0 · y. Their support sets coincide. Their residue-field fiber dimensions also coincide at every prime. But their Fitting ideals do not…
Exact mathematical passage
M=\coker\begin{pmatrix}x&0\\0&y\end{pmatrix}, \qquad M'=\coker\begin{pmatrix}x^2&0\\0&y\end{pmatrix}.
m_1Use in source · 6 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 600
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
m_1,\ldots,m_d
M_2Use in source · 3 passages
Search aliases: M_2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 397
The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles ar…
Exact mathematical passage
M_2
m_2Use in source · 5 passages
Search aliases: m_2
Use in source · trace_separator.tex, line 9
draw[->,black!35] (-2.5,0)--(2.5,0); draw[->,black!35] (0,-2.3)--(0,2.3) node[above] second coordinate; draw[->,line width=1.2pt] (0,0)--(1.5,1.5); draw[->,line width=1.2pt] (0,0)--(1.5,-1.5); node[anchor=south west] at (1.5,1.5) (1,1); node[anchor=north west] at (1.5,-1.5) (1,-1); node[anchor=west] at (3.3,.7) m_1+m_2=5; node[anchor=west] at (3.3,-.2) m_1-m_2=1; node[anchor=west,font=sffamilybfseries] at (3.3,-1.3) solution: m_1=3, m_2=2;
Exact mathematical passage
m_1+m_2=5
m_dUse in source · 1 passage
Search aliases: m_d
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 600
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
m_1,\ldots,m_d
m_iUse in source · 1 passage
Search aliases: m_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 601
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
s_im_i=0
\m_\OOUse in source · 1 passage
Search aliases: m_OO m_ O m OO m mathcal
Use in source · pages.tex, line 665
midrule Empty family · S= O and a=m_ O. The scalar adic topology remains; an empty sum of factor ideals does not erase it.
Exact mathematical passage
\mathfrak a=\m_\OO
\m_{\OO}Use in source · 1 passage
Search aliases: m_OO m_ O m OO m mathcal
Use in source · pages.tex, line 649
-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition: S=⊗_i=1^s R_i, a=m_ OS+∑_im_R_iS, L=∑_i I_iS. The comparison uses the a-adic additive uniformity on S, the image ideal topology on S/L…
Exact mathematical passage
S=\bigotimes_{i=1}^s R_i,\qquad \mathfrak a=\m_{\OO}S+\sum_i\m_{R_i}S,\qquad L=\sum_i I_iS.
\m_{R_i}Use in source · 1 passage
Search aliases: m_R_i m
Use in source · pages.tex, line 649
-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition: S=⊗_i=1^s R_i, a=m_ OS+∑_im_R_iS, L=∑_i I_iS. The comparison uses the a-adic additive uniformity on S, the image ideal topology on S/L…
Exact mathematical passage
S=\bigotimes_{i=1}^s R_i,\qquad \mathfrak a=\m_{\OO}S+\sum_i\m_{R_i}S,\qquad L=\sum_i I_iS.
\MaxUse in source · 2 passages
Search aliases: Max operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1024
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
\midruleUse in source · 1 passage
Search aliases: midrule
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
NUse in source · 11 passages
Search aliases: N
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1069
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
N\geq1
\mathbb NUse in source · 2 passages
Search aliases: N mathbb
Use in source · pages.tex, line 566
[t].44linewidthcentering sffamilybfseries Fixed carrier k^5 vfig[.75]nested_finite captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_tail captiontextRow m retains all basis vectors e_j with j≥ m. Every row continues infinitely to the right. The display is a finite window, not a trunca…
Exact mathematical passage
\ell^2(\mathbb N)
\mathfrak nUse in source · 1 passage
Search aliases: n mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1032
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
N^3Use in source · 4 passages
Search aliases: N^3
Use in source · pages.tex, line 533
0 · 0 · 0, P^2=P. vfig[.45]core_projection captiontextAt every positive horizon, precisely the e_1 direction remains visible. hfill [t].48linewidthcentering sffamilybfseries Every direction eventually dies N=0 · 1 · 0 0 · 0 · 1 0 · 0 · 0, N^3=0. vfig[.45]core_nilpotent captiontextImage dimensions are 3,2,1,0,0,…, although every level still has dimension three.
Exact mathematical passage
N=\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix},\qquad N^3=0.
n_1Use in source · 1 passage
Search aliases: n_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
n_iUse in source · 3 passages
Search aliases: n_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
n_rUse in source · 1 passage
Search aliases: n_r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
\nilUse in source · 1 passage
Search aliases: nil Nil operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 679
[Near faithfulness is full support] Let R be Noetherian and let M be a finite R module. The following conditions are equivalent. item M is nearly faithful. item Supp_R(M)=Spec R. item Ann_R(M)⊆Nil(R). If R is reduced, these conditions are equivalent to faithfulness.
Exact mathematical passage
\Ann_R(M)\subseteq\nil(R)
\nmidUse in source · 1 passage
Search aliases: nmid
Use in source · pages.tex, line 696
headWhat makes this a finite certificate The manuscript also checks absolute irreducibility and projector detection on every irreducible adjoint submodule. In this example 7nmid2, and every group element's eigenvalues lie in F_7, as required by the stated convention. The two displayed pairings alone are a worked part of that complete certificate.
Exact mathematical passage
7\nmid2
\NNUse in source · 1 passage
Search aliases: NN N mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1255
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
V=\ell^2(\NN)
OUse in source · 1 passage
Search aliases: O
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\mathcal OUse in source · 2 passages
Search aliases: O mathcal
Use in source · monomial_quotient.tex, line 45
ode[font=scriptsize] at (0.345,3.3449999999999998) y^4; draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,4.095) y^5; node[anchor=west,font=footnotesize] at (0,-.46) i+j<6 shown; coefficients lie in O.; node[anchor=west,font=sffamilysmallbfseries] at (6.35,5.10) After imposing (x^2,y); draw[fill=black!12,draw=black!35] (6.35,0.0…
Exact mathematical passage
\mathcal O
oUse in source · 1 passage
Search aliases: o
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\OO^dUse in source · 2 passages
Search aliases: OO^d O^d OO mathcal
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 528
[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.
Exact mathematical passage
\boxed{ M/\fraka M \cong M\otimes_{\Lambda,\epsilon}\OO \cong \OO^d. }
\OO_\lambdaUse in source · 10 passages
Search aliases: OO_λ O_λ OO lambda mathcal lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1465
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
\OO_\lambda
\overline{JB}Use in source · 2 passages
Search aliases: overlineJB overline
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 316
is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The construction commutes with the canonical map from B, which is therefore surjective. It is also unchanged by transporting the t…
Exact mathematical passage
R^\tau\cong B/\overline{JB}.
\overline{L\widehat S}Use in source · 3 passages
Search aliases: overlineLwidehat S overline widehat
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 284
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\pUse in source · 3 passages
Search aliases: p
Use in source · pages.tex, line 188
Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full algebraic loci over any field.
Exact mathematical passage
\p
P^2Use in source · 1 passage
Search aliases: P^2
Use in source · pages.tex, line 525
[t].48linewidthcentering sffamilybfseries A surviving direction P=1 · 0 · 0 0 · 0 · 0 0 · 0 · 0, P^2=P. vfig[.45]core_projection captiontextAt every positive horizon, precisely the e_1 direction remains visible. hfill [t].48linewidthcentering sffamilybfseries Every direction eventually dies N=0 · 1 · 0
Exact mathematical passage
P=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix},\qquad P^2=P.
p^mUse in source · 1 passage
Search aliases: p^m
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 545
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\Delta_m=(\ZZ/p^m\ZZ)^r
p^{n_1}Use in source · 1 passage
Search aliases: p^n_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
p^{n_i}Use in source · 3 passages
Search aliases: p^n_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
p^{n_r}Use in source · 1 passage
Search aliases: p^n_r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
P_{D,\alpha}Use in source · 2 passages
Search aliases: P_D,α alpha
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 433
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
P_{D,\alpha}=(\alpha-\beta)^{-1}(D-\beta I).
P_mUse in source · 5 passages
Search aliases: P_m
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1255
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
P_m
P_{s,-1}Use in source · 1 passage
Search aliases: P_s,-1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 439
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
P_{s,1}Use in source · 2 passages
Search aliases: P_s,1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 428
3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4 0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
P_{s,1}=\begin{pmatrix}0&4\\0&1\end{pmatrix}, \qquad \Tr(P_{s,1}W_{\mathrm{std}})=6\ne0.
P_{t,2}Use in source · 2 passages
Search aliases: P_t,2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 424
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4
Exact mathematical passage
P_{t,2}=\begin{pmatrix}2&4\\3&6\end{pmatrix}, \qquad \Tr(P_{t,2}W_{\mathrm{sgn}})=3\ne0,
P_{tst^{-1},1}Use in source · 1 passage
Search aliases: P_tst^-1,1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 439
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
P_{\lambda,v}Use in source · 4 passages
Search aliases: P_λ,v lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1478
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
P_{\lambda,v}\in\OO_\lambda[X]
\pagerefVUse in source · 1 passage
Search aliases: pagerefV
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\pmodUse in source · 3 passages
Search aliases: pmod
Use in source · pages.tex, line 416
headThe projectors can be written explicitly Modulo (z-1)^2(z+1)^3, put e_+(z)=frac(5-3z)(z+1)^316, e_-(z)=1-e_+(z). Then e_+≡1pmod(z-1)^2 and e_+≡0pmod(z+1)^3. Consequently e_+(H)=diag(1,1,0,0,0), e_-(H)=diag(0,0,1,1,1). The packet dimensions are two and three. The ordinary eigenspaces each have dimension one; the generalized eigenspaces have dimensions two and three.
Exact mathematical passage
e_+\equiv1\pmod{(z-1)^2}
QUse in source · 3 passages
Search aliases: Q
Use in source · pages.tex, line 757
headCalculate the mismatch For Q=(1 · 1 0 · 1), QDQ^-1=1 · -2
Exact mathematical passage
Q=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)
\mathbb QUse in source · 3 passages
Search aliases: Q mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1824
The original comparison wording did not explicitly supply a compatible Hecke point over the fixed coefficient field. Support preservation alone is insufficient: extension from Q to Q(i) preserves full support, but there is no ring map Q(i)→ Q extending the identity. The revision therefore states y∘ f=x and exposes the arithmetic comparison and existing-family inputs. Residual occupancy, representability and the applicable patching theorem remain supplied arithmetic provenance. The original full …
Exact mathematical passage
\mathbb Q
\mathfrak qUse in source · 1 passage
Search aliases: q mathfrak
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
qUse in source · 12 passages
Search aliases: q
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 473
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
q:\Delta'\to\Delta
Q^{-1}Use in source · 2 passages
Search aliases: Q^-1
Use in source · pages.tex, line 758
0 · 1), QDQ^-1=1 · -2 0 · -1 ≡1 · 23 0 · 24pmod25. It reduces to (1 · 3
Exact mathematical passage
QDQ^{-1}=\begin{pmatrix}1&-2\\0&-1\end{pmatrix} \equiv\begin{pmatrix}1&23\\0&24\end{pmatrix}\pmod{25}.
q_*Use in source · 3 passages
Search aliases: q_*
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 474
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
q_*:\OO[\Delta']\longrightarrow\OO[\Delta].
\QQUse in source · 15 passages
Search aliases: QQ Q mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 201
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
E/\QQ_p
\QQ_2Use in source · 3 passages
Search aliases: QQ_2 Q_2 QQ mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1579
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\QQ_2
\QQ_pUse in source · 1 passage
Search aliases: QQ_p Q_p QQ mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 201
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
E/\QQ_p
rUse in source · 36 passages
Search aliases: r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
r'Use in source · 1 passage
Search aliases: r'
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
R^2Use in source · 1 passage
Search aliases: R^2
Use in source · pages.tex, line 169
% --------------------------------------------------------------------------- vpage6A matrix becomes a support locusMODEL B / FITTING STRATAMain text: Theorems 6.2, 6.3 and 6.4 p12-route:visual-6Main statement 6.3 Now change models. Let R=k[x,y] for a field k, and present a module by R^2xrightarrowAR^2⟶ M⟶0, A=x · 0 0 · y. The relations are xe_1=0 and ye_2=0, so M≅ R/(x)⊕ R/(y).
Exact mathematical passage
R^2\xrightarrow{A}R^2\longrightarrow M\longrightarrow0, \qquad A=\begin{pmatrix}x&0\\0&y\end{pmatrix}.
R^mUse in source · 1 passage
Search aliases: R^m
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 619
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
R^m\xrightarrow{A}R^n\longrightarrow M\longrightarrow0
R^nUse in source · 1 passage
Search aliases: R^n
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 619
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
R^m\xrightarrow{A}R^n\longrightarrow M\longrightarrow0
R^\tauUse in source · 5 passages
Search aliases: R^τ tau
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 316
is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The construction commutes with the canonical map from B, which is therefore surjective. It is also unchanged by transporting the t…
Exact mathematical passage
R^\tau\cong B/\overline{JB}.
R_1Use in source · 6 passages
Search aliases: R_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 257
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R_1,\ldots,R_s
R_2Use in source · 3 passages
Search aliases: R_2
Use in source · pages.tex, line 66
% --------------------------------------------------------------------------- vpage3Local conditions remove directionsMODEL A / A FORMAL THICKENINGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_1/I_1…
Exact mathematical passage
R_1=\OO[[x]],\quad I_1=(x^2),\qquad R_2=\OO[[y]],\quad I_2=(y).
R_{\frakp}Use in source · 3 passages
Search aliases: R_frakp R_ p frakp mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 597
Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.
Exact mathematical passage
R_{\frakp}
R_{\frakp}^nUse in source · 1 passage
Search aliases: R_frakp^n R_ p^n frakp mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 623
from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
R_{\frakp}^n
R_sUse in source · 3 passages
Search aliases: R_s
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 257
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R_1,\ldots,R_s
\raggedrightUse in source · 1 passage
Search aliases: raggedright
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\rankUse in source · 3 passages
Search aliases: rank operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 649
Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equivalent to the vanishing of every minor of size n-r+1. Those minors generate Fitt_r-1^R(M).
Exact mathematical passage
n-\rank A(\frakp).
\rank_{E'}Use in source · 1 passage
Search aliases: rank_E' rank operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1092
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\RRUse in source · 1 passage
Search aliases: RR
Use in source · pages.tex, line 194
Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full algebraic loci over any field.
Exact mathematical passage
k=\RR
s^2Use in source · 1 passage
Search aliases: s^2
Use in source · pages.tex, line 682
1 · 6. All entries and calculations on this page are modulo seven. Direct multiplication gives s^2=t^3=I and sts=t^-1. Their eigenvalues are {-1,1} and {2,4}. Distinct eigenvalues allow an exact projector formula P_D,α=(α-β)^-1(D-β I).
Exact mathematical passage
s^2=t^3=I
s_1Use in source · 1 passage
Search aliases: s_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 602
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
s=s_1\cdots s_d
S_3Use in source · 14 passages
Search aliases: S_3
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 355
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
G=S_3
s_dUse in source · 1 passage
Search aliases: s_d
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 602
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
s=s_1\cdots s_d
s_iUse in source · 2 passages
Search aliases: s_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 601
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
s_i\notin\frakp
\mathrm{sgn}Use in source · 2 passages
Search aliases: sgn mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 422
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{sgn}}=Z
\operatorname{sgn}Use in source · 3 passages
Search aliases: sgn operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 386
[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.
Exact mathematical passage
\ad^0V\cong\operatorname{sgn}\oplus V.
\smallUse in source · 1 passage
Search aliases: small
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1555
Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.
Exact mathematical passage
W = \operatorname{span} \left\{ (t_1(\omega),\ldots,t_r(\omega)):\omega\in\Omega \right\} \subseteq E'^r.
\operatorname{Spec}Use in source · 2 passages
Search aliases: Spec operatorname
Use in source · node_spectrum.tex, line 3
node[font=sffamilysmallbfseries] at (1.6,3.15) SpecR; draw[->,line width=1.1pt] (.1,2.1)--(1.6,.35); draw[->,dashed] (3.1,2.1)--(1.6,.35); fill (.1,2.1) circle (.09);draw[fill=white] (3.1,2.1) circle (.09);fill (1.6,.35) circle (.09); node[above=6pt] at (.1,2.1) (x);node[above=6pt] at (3.1,2.1) (y); node[below=6pt] at (1.6,.35) (x,y); node[font=footnotesize,align=center] at (1.6,-.7) Filled points: support of R/(x).
Exact mathematical passage
\operatorname{Spec}R
\mathrm{ss}Use in source · 6 passages
Search aliases: ss mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1361
tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\boxed{ \rho_x^{\mathrm{ss}} \cong \rho_{\pi_x}. }
\mathrm{std}Use in source · 2 passages
Search aliases: std mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 423
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{std}}=W(6,4,3)=3A+3B
\Supp_{R/I}Use in source · 2 passages
Search aliases: Supp_R/I Supp operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 706
[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).
Exact mathematical passage
\boxed{ \Supp_{R/I}(M/IM) = \Supp_R(M)\cap V(I). }
\Supp_{R_\infty/(t)}Use in source · 1 passage
Search aliases: Supp_R_∞/(t) Supp infty operatorname infty
Use in source · pages.tex, line 347
headThe actual descent theorem is one-way For a finite module over a Noetherian ring, Supp_R_∞/(t)(M_∞/tM_∞) =Supp_R_∞(M_∞)∩ V(t). If the initial support is full, the resulting support is full on the quotient. Our example shows that full support at the quotient can occur even when initial support was not full. It also shows why freeness over S alone is not the required patching hypothesis.
Exact mathematical passage
\Supp_{R_\infty/(t)}(M_\infty/tM_\infty) =\Supp_{R_\infty}(M_\infty)\cap V(t).
\Supp_TUse in source · 1 passage
Search aliases: Supp_T Supp operatorname
Use in source · pages.tex, line 431
emptyMODEL C / OCCUPANCY IS MODULE DATAMain text: Theorem 7.4, Theorem 8.2 and Corollary 8.4 p12-route:visual-13Main statement 7.4 Keep the algebra T=T_+× T_- from pagerefV12, but compare two different T-modules: V=T, W=T_+, where T acts on W through projection to its first factor. The algebra still has two maximal ideals, but only one of them is in Supp_T(W). Here T is a supplied algebra; its action on W is not assumed faithful.
Exact mathematical passage
\Supp_T(W)
TUse in source · 28 passages
Search aliases: T
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1063
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
T_1,\ldots,T_d
tUse in source · 56 passages
Search aliases: t
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 362
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1 0 · 1 , t= 0 · 6 1 · 6 .
Exact mathematical passage
s= \begin{pmatrix} 6&1\\ 0&1 \end{pmatrix}, \qquad t= \begin{pmatrix} 0&6\\ 1&6 \end{pmatrix}.
t^{-1}Use in source · 2 passages
Search aliases: t^-1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 439
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
t^2Use in source · 3 passages
Search aliases: t^2
Use in source · pages.tex, line 271
[c].44linewidth vfig[.94]diagonal_slice hfill [c].51linewidth headBoth relations vanish together (xy)⟼(t^2), (x,y)⟼(t). The support is the origin, but the origin has fiber dimension two. Replacing (t^2) by its radical (t) would lose the zeroth Fitting ideal.
Exact mathematical passage
(xy)\longmapsto(t^2),\qquad (x,y)\longmapsto(t).
t^3Use in source · 1 passage
Search aliases: t^3
Use in source · pages.tex, line 682
1 · 6. All entries and calculations on this page are modulo seven. Direct multiplication gives s^2=t^3=I and sts=t^-1. Their eigenvalues are {-1,1} and {2,4}. Distinct eigenvalues allow an exact projector formula P_D,α=(α-β)^-1(D-β I).
Exact mathematical passage
s^2=t^3=I
T_+Use in source · 6 passages
Search aliases: T_+
Use in source · pages.tex, line 395
% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ underbrace Q[ε]/(ε^2)_T_+, z=1+ε × underbrace Q[η]/(η^3)_T_-, z=-1+η. Multiplication by z is an operator H. In the ordered local bases 1,ε and 1,η,η^2, it…
Exact mathematical passage
T\cong \underbrace{\QQ[\varepsilon]/(\varepsilon^2)}_{T_+,\ z=1+\varepsilon} \times \underbrace{\QQ[\eta]/(\eta^3)}_{T_-,\ z=-1+\eta}.
T_-Use in source · 3 passages
Search aliases: T_-
Use in source · pages.tex, line 395
% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ underbrace Q[ε]/(ε^2)_T_+, z=1+ε × underbrace Q[η]/(η^3)_T_-, z=-1+η. Multiplication by z is an operator H. In the ordered local bases 1,ε and 1,η,η^2, it…
Exact mathematical passage
T\cong \underbrace{\QQ[\varepsilon]/(\varepsilon^2)}_{T_+,\ z=1+\varepsilon} \times \underbrace{\QQ[\eta]/(\eta^3)}_{T_-,\ z=-1+\eta}.
T_1Use in source · 2 passages
Search aliases: T_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1063
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
T_1,\ldots,T_d
t_1Use in source · 4 passages
Search aliases: t_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1545
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
t_1,\ldots,t_r
t_2Use in source · 1 passage
Search aliases: t_2
Use in source · pages.tex, line 806
% --------------------------------------------------------------------------- vpage25Recover multiplicities from tracesA FINITE SEPARATOR / A KNOWN SEMISIMPLE CARRIERMain text: Proposition 10.9 p12-route:visual-25Main statement 10.9 Suppose a known semisimple carrier is a sum of two known constituents with multiplicities m_1,m_2. Consider the independent trace functions t_1(ω)=1, t_2(ω)=ω, ω∈ Q. At the two sample points ω=1,-1, their evaluation matrix is A=1 · 1
Exact mathematical passage
t_1(\omega)=1,\qquad t_2(\omega)=\omega,\qquad \omega\in\QQ.
T_dUse in source · 2 passages
Search aliases: T_d
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1063
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
T_1,\ldots,T_d
T_iUse in source · 5 passages
Search aliases: T_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1070
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
\boxed{ e_{\frakm}V = \left\{ v\in V: (T_i-\lambda_i)^Nv=0 \text{ for every }i \right\}. }
t_iUse in source · 1 passage
Search aliases: t_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1547
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\left(t_i(\omega_j)\right)_{j,i}
t_rUse in source · 3 passages
Search aliases: t_r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1545
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
t_1,\ldots,t_r
\takeawayUse in source · 1 passage
Search aliases: takeaway
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\topruleUse in source · 1 passage
Search aliases: toprule
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\TrUse in source · 3 passages
Search aliases: Tr operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 424
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4
Exact mathematical passage
P_{t,2}=\begin{pmatrix}2&4\\3&6\end{pmatrix}, \qquad \Tr(P_{t,2}W_{\mathrm{sgn}})=3\ne0,
\operatorname{Tr}Use in source · 2 passages
Search aliases: Tr operatorname
Use in source · adequacy_pairings.tex, line 6
3 · 6; node[anchor=west] at (8.2,1.8) Tr(PW)=3; draw[black!30] (0,.9)--(11.3,.9); node[anchor=west,font=sffamilybfseries] at (0,.3) standard constituent; node[anchor=west] at (0,-.7) W=6 · 4
Exact mathematical passage
\operatorname{Tr}(PW)=3
\Tr_{E'}Use in source · 1 passage
Search aliases: Tr_E' Tr operatorname
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1092
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\TT_{\frakm}Use in source · 3 passages
Search aliases: TT_frakm T_ m TT frakm mathbb mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1024
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
UUse in source · 3 passages
Search aliases: U
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1380
[Full support gives pointwise automorphy on the realized locus] In the setting of [source reference], suppose Supp_R(H)=Spec R. For every characteristic zero point x:R→ E' with ker x∈ U⊆Spec R, assume the coefficient topology and continuity hypotheses of [source reference] and the declared classicality and Galois realization input. Then every such point is automorphic in the precise sense of [source reference].
Exact mathematical passage
\ker x\in U\subseteq\Spec R
uUse in source · 22 passages
Search aliases: u
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 404
A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomology vanishings.
Exact mathematical passage
u=-6^{-1}\sum_{h\in S_3}f(h).
u^2Use in source · 2 passages
Search aliases: u^2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1128
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
Exact mathematical passage
\TT = E'[u]/(u^2) \times E'[v]/(v^3)
v^3Use in source · 1 passage
Search aliases: v^3
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1128
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
Exact mathematical passage
\TT = E'[u]/(u^2) \times E'[v]/(v^3)
v_1Use in source · 1 passage
Search aliases: v_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 356
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
v_1=(1,-1,0), \qquad v_2=(0,1,-1),
v_2Use in source · 1 passage
Search aliases: v_2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 356
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
v_1=(1,-1,0), \qquad v_2=(0,1,-1),
V_{\frakm}Use in source · 2 passages
Search aliases: V_frakm V_ m frakm mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1051
[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).
Exact mathematical passage
\boxed{ e_{\frakm}V\cong V_{\frakm}.}
v_mUse in source · 2 passages
Search aliases: v_m
Use in source · pages.tex, line 521
% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions visible at level m after looking h levels ahead form im(F^h).
Exact mathematical passage
v_m=Fv_{m+1}
V_{m+1}Use in source · 3 passages
Search aliases: V_m+1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1204
[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.
Exact mathematical passage
V_{m+1}\to V_m
v_{m+1}Use in source · 1 passage
Search aliases: v_m+1
Use in source · pages.tex, line 521
% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions visible at level m after looking h levels ahead form im(F^h).
Exact mathematical passage
v_m=Fv_{m+1}
v_{m+3}Use in source · 1 passage
Search aliases: v_m+3
Use in source · pages.tex, line 542
headNow compute the limits, not just their level dimensions For the P-system, the limit consists of the constant compatible sequences (a e_1,a e_1,…), so it is isomorphic to k. For the N-system, every compatible sequence satisfies v_m=N^3v_m+3=0 for every m, and the limit is zero. This is an all-level argument from the identity N^3=0, not a finite experimental guess.
Exact mathematical passage
v_m=N^3v_{m+3}=0\quad\text{for every }m,
V_uUse in source · 2 passages
Search aliases: V_u
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1136
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
Exact mathematical passage
V=V_u\oplus V_v
V_vUse in source · 2 passages
Search aliases: V_v
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1136
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
Exact mathematical passage
V=V_u\oplus V_v
\varprojlimUse in source · 3 passages
Search aliases: varprojlim
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1232
[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.
Exact mathematical passage
\boxed{ \varprojlim_mV_m \neq0. }
\varprojlim_mUse in source · 2 passages
Search aliases: varprojlim_m varprojlim
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1232
[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.
Exact mathematical passage
\boxed{ \varprojlim_mV_m \neq0. }
\varprojlim_nUse in source · 1 passage
Search aliases: varprojlim_n varprojlim
Use in source · pages.tex, line 131
% --------------------------------------------------------------------------- vpage5Completion retains every precisionMODEL A / COMPATIBLE COEFFICIENTSMain text: Theorem 3.1; reconstruction and finite occupancy in Section 10 p12-route:visual-5Main statement 10.5 For B= Z_5[ε]/(ε^2), consider the finite quotient rings B_n=( Z/5^n Z)[ε]/(ε^2), |B_n|=5^2n, B≅varprojlim_n B_n. A point of the limit is a single compatible choice at every level, not unrelated choices from the nonempty finite rings.
Exact mathematical passage
B_n=(\ZZ/5^n\ZZ)[\varepsilon]/(\varepsilon^2), \qquad |B_n|=5^{2n},\qquad B\cong\varprojlim_n B_n.
\vpageUse in source · 1 passage
Search aliases: vpage
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
Exact mathematical passage
3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
wUse in source · 3 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 521
A vector in the regular representation fixed by all left translations has constant coefficients. Consequently ve=ϵ(e)v=0. On the other hand, e^2=e implies we=w for every w∈ W, hence ve=v, a contradiction. Thus every idempotent of augmentation zero is zero. Since ϵ(e)^2=ϵ(e) in the field k, either ϵ(e)=0 or ϵ(e)=1; in the latter case apply the preceding argument to 1-e.
Exact mathematical passage
we=w
W_{\mathrm{sgn}}Use in source · 2 passages
Search aliases: W_sgn mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 422
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
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W_{\mathrm{sgn}}=Z
W_{\mathrm{std}}Use in source · 2 passages
Search aliases: W_std mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 423
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{std}}=W(6,4,3)=3A+3B
\whotUse in source · 3 passages
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Use in source · pages.tex, line 71
NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_1/I_1)whot_ O(R_2/I_2) ≅ A/J ≅ O[ε]/(ε^2).
Exact mathematical passage
(R_1/I_1)\whot_{\OO}(R_2/I_2) \cong A/J \cong \OO[\varepsilon]/(\varepsilon^2).
\whot_AUse in source · 1 passage
Search aliases: whot_A whot
Use in source · pages.tex, line 671
For global base change, the structural map A→ B belongs to the continuous local coefficient-algebra setting. Then Bwhot_A(A/J)≅ B/overlineJB. For a continuous algebra point to a complete Hausdorff target, closedness of its kernel gives the factorization test. Locality of a point, when supplied, passes to its unique quotient factor.
Exact mathematical passage
B\whot_A(A/J)\cong B/\overline{JB}
\whot_{\OO}Use in source · 1 passage
Search aliases: whot_OO whot_ O whot OO whot mathcal
Use in source · pages.tex, line 71
NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_1/I_1)whot_ O(R_2/I_2) ≅ A/J ≅ O[ε]/(ε^2).
Exact mathematical passage
(R_1/I_1)\whot_{\OO}(R_2/I_2) \cong A/J \cong \OO[\varepsilon]/(\varepsilon^2).
\widehat SUse in source · 5 passages
Search aliases: widehat S widehat
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 284
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\widehat{S/L}Use in source · 2 passages
Search aliases: widehatS/L widehat
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 284
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\wideotimes_{\OO}Use in source · 2 passages
Search aliases: wideotimes_OO mathbinwidehat⊗_ O wideotimes OO mathbin widehat otimes mathcal
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 258
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R^{\mathrm{loc}}=R_1\wideotimes_{\OO}\cdots\wideotimes_{\OO}R_s
X^2Use in source · 2 passages
Search aliases: X^2
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1588
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ \FF_2[C_2] \cong \FF_2[X]/(X^2), \qquad X=g-1. }
X^{2}Use in source · 2 passages
Search aliases: X^2
Use in source · augmentation_grid.tex, line 7
[x=1cm,y=1cm,font=small,>=Stealth] draw[black!40,fill=black!16] (0.0,0.0) rectangle ++(1,.9); node at (0.5,0.45) 1; draw[black!40,fill=white] (1.2,0.0) rectangle ++(1,.9); node at (1.7,0.45) X; draw[black!40,fill=white] (2.4,0.0) rectangle ++(1,.9); node at (2.9,0.45) X^2; draw[black!40,fill=white] (3.5999999999999996,0.0) rectangle ++(1,.9); node at (4.1,0.45) X^3; draw[black!40,fill=white] (0.0,1.1) rectangle ++(1,.9); node at (0.5,1.55) Y; draw[black!40,fill=white] (1.2,1.1) rectangle ++(1,.9…
Exact mathematical passage
X^{2}
x^2Use in source · 15 passages
Search aliases: x^2
Use in source · monomial_quotient.tex, line 46
0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,4.095) y^5; node[anchor=west,font=footnotesize] at (0,-.46) i+j<6 shown; coefficients lie in O.; node[anchor=west,font=sffamilysmallbfseries] at (6.35,5.10) After imposing (x^2,y); draw[fill=black!12,draw=black!35] (6.35,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (6.694999999999999,0.345) 1; draw[fill…
Exact mathematical passage
(x^2,y)
x^{2}Use in source · 4 passages
Search aliases: x^2
Use in source · monomial_quotient.tex, line 8
familysmallbfseries] at (0,5.10) Before imposing the ideals; draw[fill=white,draw=black!35] (0.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,0.345) 1; draw[fill=white,draw=black!35] (0.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x…
Exact mathematical passage
x^{2}
X^{3}Use in source · 2 passages
Search aliases: X^3
Use in source · augmentation_grid.tex, line 9
ikzpicture[x=1cm,y=1cm,font=small,>=Stealth] draw[black!40,fill=black!16] (0.0,0.0) rectangle ++(1,.9); node at (0.5,0.45) 1; draw[black!40,fill=white] (1.2,0.0) rectangle ++(1,.9); node at (1.7,0.45) X; draw[black!40,fill=white] (2.4,0.0) rectangle ++(1,.9); node at (2.9,0.45) X^2; draw[black!40,fill=white] (3.5999999999999996,0.0) rectangle ++(1,.9); node at (4.1,0.45) X^3; draw[black!40,fill=white] (0.0,1.1) rectangle ++(1,.9); node at (0.5,1.55) Y; draw[black!40,fill=white] (1.2,1.1) rectang…
Exact mathematical passage
X^{3}
x^{3}Use in source · 3 passages
Search aliases: x^3
Use in source · monomial_quotient.tex, line 10
node[font=scriptsize] at (0.345,0.345) 1; draw[fill=white,draw=black!35] (0.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x^3; draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,0.345) x^4; …
Exact mathematical passage
x^{3}
X^4Use in source · 2 passages
Search aliases: X^4
Use in source · pages.tex, line 703
-------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-1. Then F_2[Δ]≅ F_2[X,Y]/(X^4,Y^2). The identity (1+X)^4-1=X^4 and its Y analogue provide the relations. The eight surviving monomials provide the exact …
Exact mathematical passage
\FF_2[\Delta]\cong\FF_2[X,Y]/(X^4,Y^2).
x^{4}Use in source · 2 passages
Search aliases: x^4
Use in source · monomial_quotient.tex, line 12
tsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x^3; draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,0.345) x^4; draw[fill=white,draw=black!35] (3.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (4.095,0.345) x^5; draw[fill=whi…
Exact mathematical passage
x^{4}
x^{5}Use in source · 1 passage
Search aliases: x^5
Use in source · monomial_quotient.tex, line 14
] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x^3; draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,0.345) x^4; draw[fill=white,draw=black!35] (3.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (4.095,0.345) x^5; draw[fill=white,draw=black!35] (0.0,0.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,1.095) y; draw[fill=white,d…
Exact mathematical passage
x^{5}
x^iUse in source · 1 passage
Search aliases: x^i
Use in source · pages.tex, line 81
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
Exact mathematical passage
f=\sum_{i,j\ge0}a_{ij}x^iy^j
X_1Use in source · 4 passages
Search aliases: X_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
X_1^{p^{n_1}}Use in source · 1 passage
Search aliases: X_1^p^n_1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
X_2Use in source · 3 passages
Search aliases: X_2
Use in source · finite_forest.tex, line 11
) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,1.6) X_3: two admissible nodes; node[anchor=west,font=footn…
Exact mathematical passage
X_2
X_3Use in source · 4 passages
Search aliases: X_3
Use in source · finite_forest.tex, line 12
h=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,1.6) X_3: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,.4) X_n: one node for every n≥4;
Exact mathematical passage
X_3
X_4Use in source · 1 passage
Search aliases: X_4
Use in source · pages.tex, line 740
headWhat the compactness argument proves For general finite levels, the reduction forest is a locally finite rooted tree with vertices at every depth. An infinite branch exists. Surjectivity is sufficient but is not necessary: the map X_3→ X_2 above misses b, and X_4→ X_3 misses d.
Exact mathematical passage
X_4\to X_3
x_{g,a,b}Use in source · 2 passages
Search aliases: x_g,a,b
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1485
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
Exact mathematical passage
B_\lambda=\OO_\lambda[x_{g,a,b}:g\in\Gamma,\ a,b\in D].
X_iUse in source · 7 passages
Search aliases: X_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 493
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
X_i=\gamma_i-1
X_i^{p^{n_i}}Use in source · 1 passage
Search aliases: X_i^p^n_i
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 508
In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique maximal ideal.
Exact mathematical passage
(1+X_i)^{p^{n_i}}-1=X_i^{p^{n_i}}.
X_nUse in source · 2 passages
Search aliases: X_n
Use in source · finite_forest.tex, line 13
draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,1.6) X_3: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,.4) X_n: one node for every n≥4;
Exact mathematical passage
X_n
X_rUse in source · 2 passages
Search aliases: X_r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
X_r^{p^{n_r}}Use in source · 1 passage
Search aliases: X_r^p^n_r
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
X_{\lambda,m}Use in source · 3 passages
Search aliases: X_λ,m lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1442
[Companion branch criterion] Suppose every level X_λ,m is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then varprojlim_mX_λ,m ≠∅.
Exact mathematical passage
X_{\lambda,m}
X_{\lambda,n}Use in source · 3 passages
Search aliases: X_λ,n lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1472
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
X_{\lambda,n}
\xrightarrowUse in source · 6 passages
Search aliases: xrightarrow
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 619
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
R^m\xrightarrow{A}R^n\longrightarrow M\longrightarrow0
yUse in source · 120 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 960
[A quotient can erase a component] Let R=k[[x,y]]/(xy) and let T=R/(x)≅ k[[y]]. Viewed as an R module, T has support V(x), which misses the generic point (y) of the other irreducible component. Thus a surjection R↠ T does not by itself imply full component occupancy.
Exact mathematical passage
R=k[[x,y]]/(xy)
Y^2Use in source · 1 passage
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Use in source · pages.tex, line 703
-------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-1. Then F_2[Δ]≅ F_2[X,Y]/(X^4,Y^2). The identity (1+X)^4-1=X^4 and its Y analogue provide the relations. The eight surviving monomials provide the exact …
Exact mathematical passage
\FF_2[\Delta]\cong\FF_2[X,Y]/(X^4,Y^2).
y^{2}Use in source · 4 passages
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Use in source · monomial_quotient.tex, line 26
(1.845,1.095) x^2y; draw[fill=white,draw=black!35] (2.25,0.75) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,1.095) x^3y; draw[fill=white,draw=black!35] (3.0,0.75) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,1.095) x^4y; draw[fill=white,draw=black!35] (0.0,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,1.845) y^2; draw[fill=white,draw=black!35] (0.75,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,1.845) xy^2; draw[fill=white…
Exact mathematical passage
y^{2}
y^{3}Use in source · 3 passages
Search aliases: y^3
Use in source · monomial_quotient.tex, line 34
,1.845) xy^2; draw[fill=white,draw=black!35] (1.5,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,1.845) x^2y^2; draw[fill=white,draw=black!35] (2.25,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,1.845) x^3y^2; draw[fill=white,draw=black!35] (0.0,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,2.5949999999999998) y^3; draw[fill=white,draw=black!35] (0.75,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,2.5949999999999998) xy^3; draw…
Exact mathematical passage
y^{3}
y^{4}Use in source · 2 passages
Search aliases: y^4
Use in source · monomial_quotient.tex, line 40
y^3; draw[fill=white,draw=black!35] (0.75,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,2.5949999999999998) xy^3; draw[fill=white,draw=black!35] (1.5,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,2.5949999999999998) x^2y^3; draw[fill=white,draw=black!35] (0.0,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,3.3449999999999998) y^4; draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; dr…
Exact mathematical passage
y^{4}
y^{5}Use in source · 1 passage
Search aliases: y^5
Use in source · monomial_quotient.tex, line 44
99999999998) x^2y^3; draw[fill=white,draw=black!35] (0.0,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,3.3449999999999998) y^4; draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,4.095) y^5; node[anchor=west,font=footnotesize] at (0,-.46) i+j<6 shown; coefficients lie in O.; node[anchor=west,font=sffamilysmallbfseri…
Exact mathematical passage
y^{5}
y^jUse in source · 1 passage
Search aliases: y^j
Use in source · pages.tex, line 81
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
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f=\sum_{i,j\ge0}a_{ij}x^iy^j
Y_1Use in source · 3 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1238
[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.
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Y_1
y_1Use in source · 2 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1242
Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.
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0\neq y_1\in Y_1
y_mUse in source · 3 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1243
Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.
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y_m
Y_{m+1}Use in source · 4 passages
Search aliases: Y_m+1
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1205
[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.
Exact mathematical passage
\boxed{ Y_{m+1}\twoheadrightarrow Y_m. }
y_{m+1}Use in source · 1 passage
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1243
Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.
Exact mathematical passage
y_{m+1}\in Y_{m+1}
ZUse in source · 5 passages
Search aliases: Z
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 410
Write W(a,b,c)=a · b c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.
Exact mathematical passage
W(a,b,c)=\begin{pmatrix}a&b\\c&-a\end{pmatrix}, \qquad Z=W(2,3,4),\quad A=W(1,1,2),\quad B=W(1,5,6).
\mathbb ZUse in source · 2 passages
Search aliases: Z mathbb
Use in source · polynomial_effectivity.tex, line 3
node (B) at (0,2.2) B;node (O) at (6.5,2.2) Z_5; node (BJ) at (0,0) B/J;node (On) at (6.5,0) Z/5^n Z; draw[->] (B)--(O) node[midway,above] evaluation at A; draw[->] (B)--(BJ) node[midway,left] quotient; draw[->] (O)--(On) node[midway,right] reduction; draw[->] (BJ)--(On) node[midway,below] evaluation at A_n; draw[->,dashed] (BJ)--(O) node[pos=.4,above,sloped,font=footnotesize] unique factor;
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\mathbb Z_5
zUse in source · 16 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 420
c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.
Exact mathematical passage
(z,(a,b))\mapsto zZ+aA+bB
\mathbb Z_5Use in source · 1 passage
Search aliases: Z_5 mathbb
Use in source · polynomial_effectivity.tex, line 3
node (B) at (0,2.2) B;node (O) at (6.5,2.2) Z_5; node (BJ) at (0,0) B/J;node (On) at (6.5,0) Z/5^n Z; draw[->] (B)--(O) node[midway,above] evaluation at A; draw[->] (B)--(BJ) node[midway,left] quotient; draw[->] (O)--(On) node[midway,right] reduction; draw[->] (BJ)--(On) node[midway,below] evaluation at A_n; draw[->,dashed] (BJ)--(O) node[pos=.4,above,sloped,font=footnotesize] unique factor;
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\mathbb Z_5
\ZZUse in source · 24 passages
Search aliases: ZZ Z mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 545
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\Delta_m=(\ZZ/p^m\ZZ)^r
\ZZ_2Use in source · 6 passages
Search aliases: ZZ_2 Z_2 ZZ mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1587
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\ZZ_2[C_2]
\ZZ_2^rUse in source · 2 passages
Search aliases: ZZ_2^r Z_2^r ZZ mathbb
Use in source · pages.tex, line 510
headRecovery without averaging The augmentation map sends g to 1 and has kernel (g-1). Thus Z_2[C_2]/(g-1)≅ Z_2, ( Z_2[C_2]^r)/(g-1) Z_2[C_2]^r≅ Z_2^r. These quotient identities require no division by two. They explain why augmentation and coinvariants remain usable even when sign-projector formulas do not.
Exact mathematical passage
\ZZ_2[C_2]/(g-1)\cong\ZZ_2, \qquad \bigl(\ZZ_2[C_2]^r\bigr)/(g-1)\ZZ_2[C_2]^r\cong\ZZ_2^r.
\ZZ_5Use in source · 14 passages
Search aliases: ZZ_5 Z_5 ZZ mathbb
Use in source · pages.tex, line 8
[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What it lets you see · Lessons A formal thickening · Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. · …
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3pt] \textbf{Support} & At which prime ideals does the supplied module or complex survive?\\[3pt] \textbf{Realization} & Does the surviving eigensystem come from a classical automorphic object with the required attached representation? \end{tabularx} \end{insight} \head{Three models, rather than one overloaded analogy} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Y>{\raggedright\arraybackslash}p{.15\linewidth}@{}}\toprule \textbf{Model} & \textbf{What it lets you see} & \textbf{Lessons}\\\midrule A formal thickening & Conditions on power series, infinitesimal points, obstructed lifts, and compatible precision levels. & \pagerefV{3}--\pagerefV{5}\\ A presented module & Matrix rank, support, Fitting ideals, changes of coordinates, and nonreduced structure. & \pagerefV{6}--\pagerefV{8}\\ A finite Artin algebra & Generalized eigenspaces, nilpotent directions, canonical packets, and empty packet modules. & \pagerefV{12}--\pagerefV{14}\\\bottomrule \end{tabularx} The models are connected by the paper's theorems, but they are not the same ring. A polynomial coordinate plane is not a complete local spectrum, and a local branch is not an Artin product factor. Explicitly changing models prevents a picture from silently changing the hypotheses. \head{A route through the guide} Use the object atlas on \pagerefV{2} when a symbol becomes unfamiliar. The local model starts on \pagerefV{3}; the matrix model on \pagerefV{6}. Lessons \pagerefV{9}--\pagerefV{11} explain what quotients, patching, and derived fibers preserve. Lessons \pagerefV{15}--\pagerefV{16} separate finite occupancy from genuine infinite compatibility. The exact arithmetic boundary is on \pagerefV{17}. \head{How to read a visual claim} Each lesson gives an explicit object, a calculation, and a source reference. Coordinate sketches display only the stated real points. Coefficient arrays and finite specialization diagrams describe the stated algebraic data exactly. Line thickness never stands for nilpotent multiplicity. \takeaway{These are explanations of the formal release, not additional assumptions or new Lean certificates. The accompanying scripts check the explicit finite calculations. They do not establish universal arithmetic existence.} % --------------------------------------------------------------------------- \vpage{2}{An atlas of the objects}{ORIENTATION}{Main text: Sections 2, 3, 6, 7, 8 and 12\quad \href{p12-route:visual-2}{Main statement 3.3}} A \emph{ring} describes possible points. A \emph{module over the ring} describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point. {\small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}Yp{.22\linewidth}@{}}\toprule \textbf{Symbol} & \textbf{Object and role} & \textbf{Keep separate}\\\midrule $\OO,\ k$ & Complete coefficient ring and its residue field. The formal teaching model uses $\OO=\ZZ_5$, $k=\FF_5$. & Coefficients versus variables.\\ $R_i,\ I_i$ & A supplied local ring and the ideal imposing a declared local condition. & A quotient condition is not an idempotent.\\ $R^{\mathrm{loc}}$ & Completed tensor product of the local rings over $\OO$. & Completion uses the declared adic topology.\\ $J$ & Closed ideal generated by the images of all $I_i$ in $R^{\mathrm{loc}}$. & Closure, not just a finite list of tests.\\ $R^{\mathrm{loc},\tau}$ & Completed tensor of the local quotients; identified with $R^{\mathrm{loc}}/J$. & A constructed comparison, not assumed notation.\\ $R^\square$ & Supplied global ring receiving the local structural maps. & Representability is arithmetic input.\\ $R^\tau$ & Completed global base change imposing the simultaneous local conditions. & Does not create a missing residual point.\\ $R_\infty,\ M_\infty$ & Patched ring and its module; a perfect complex is used in the derived version. & Freeness over an auxiliary ring is weaker than full support.\\ $R,\ M$ & Finite-level ring and carrier after the specified quotient and control isomorphism. & The module may be zero on a component.\\ $T,\ H$ & Hecke algebra and its ordinary or derived carrier. & Ring comparison does not imply classicality.\\ $e_{\m}H$ & Packet selected by a canonical Artin idempotent. & A generalized packet need not be an eigenspace.\\ $Y_m$ & Stable image at level $m$ of an inverse system. & Nonzero levels alone do not give a nonzero limit.\\\bottomrule \end{tabularx}} \head{Why arrows reverse when points appear} A ring map $f:R\to T$ sends a prime of $T$ to its inverse image in $R$: \[ f^*:\Spec T\longrightarrow\Spec R,\qquad \mathfrak q\longmapsto f^{-1}(\mathfrak q).
\ZZ_p^rUse in source · 1 passage
Search aliases: ZZ_p^r Z_p^r ZZ mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 546
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\OO[[\ZZ_p^r]]\cong\OO[[X_1,\ldots,X_r]],
|Use in source · 6 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1092
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
|_{y=1}Use in source · 1 passage
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Use in source · pages.tex, line 231
headMagnify transversely, rather than draw a thicker line Take the slice y=1. The second summand vanishes because y becomes a unit. What remains is M|_y=1≅ k[x]/(x), M'|_y=1≅ k[x]/(x^2). The first has basis 1; the second has basis 1,x. Multiplication by x distinguishes them exactly:
Exact mathematical passage
M|_{y=1}\cong k[x]/(x),\qquad M'|_{y=1}\cong k[x]/(x^2).
\timesUse in source · 71 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1128
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
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\TT = E'[u]/(u^2) \times E'[v]/(v^3)
\alphaUse in source · 8 passages
Search aliases: α alpha
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 432
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
(D-\alpha I)(D-\beta I)=0
\alpha_jUse in source · 3 passages
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Use in source · adic_digits.tex, line 2
[x=1cm,y=1cm,font=small,>=Stealth] node[anchor=east] at (-.18,1.1) α_j; draw[black!35,fill=black!8] (0.0,0.76) rectangle ++(.80,.67); node at (0.4,1.095) 1; draw[black!35,fill=black!12] (0.88,0.76) rectangle ++(.80,.67); node at (1.28,1.095) 2; draw[black!35,fill=black!4] (1.76,0.76) rectangle ++(.80,.67); node at (2.16,1.095) 0; draw[black!35,fill=black!4] (2.64,0.76) rectangle ++(.80,.67); node at (3.04,1.095) 0; draw[black!35,fill=black!12] (3.52,0.76) rectangle ++(.80,.67); node at (3.92,1.0…
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\alpha_j
\betaUse in source · 8 passages
Search aliases: β beta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 432
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
(D-\alpha I)(D-\beta I)=0
\beta_jUse in source · 3 passages
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Use in source · adic_digits.tex, line 23
(5.28,0.76) rectangle ++(.80,.67); node at (5.680000000000001,1.095) 2; draw[black!35,fill=black!4] (6.16,0.76) rectangle ++(.80,.67); node at (6.5600000000000005,1.095) 0; draw[black!35,fill=black!4] (7.04,0.76) rectangle ++(.80,.67); node at (7.44,1.095) 0; draw[black!35,fill=black!12] (7.92,0.76) rectangle ++(.80,.67); node at (8.32,1.095) 2; node[anchor=east] at (-.18,0.34) β_j; draw[black!35,fill=black!8] (0.0,0.0) rectangle ++(.80,.67); node at (0.4,0.335) 1; draw[black!35,fill=black!16] (…
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\beta_j
\GammaUse in source · 7 passages
Search aliases: Γ Gamma
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1465
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
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\Gamma
\gammaUse in source · 6 passages
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Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 493
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
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\gamma_i
\gamma_iUse in source · 3 passages
Search aliases: γ_i gamma
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 493
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\gamma_i
\gamma_vUse in source · 3 passages
Search aliases: γ_v gamma
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1478
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
\gamma_v\in\Gamma
\DeltaUse in source · 22 passages
Search aliases: Δ Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 460
Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.
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\Delta
\deltaUse in source · 13 passages
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Use in source · pages.tex, line 100
ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Lon…
Exact mathematical passage
D_3=\FF_5[\delta]/(\delta^3)
\Delta'Use in source · 3 passages
Search aliases: Δ' Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 473
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
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q:\Delta'\to\Delta
\delta^2Use in source · 5 passages
Search aliases: δ^2 delta
Use in source · pages.tex, line 101
d p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Longleftrightarrow a=c=0, d=b.
Exact mathematical passage
u\longmapsto a\delta+b\delta^2,\qquad v\longmapsto c\delta+d\delta^2, \qquad a,b,c,d\in\FF_5.
\delta^3Use in source · 1 passage
Search aliases: δ^3 delta
Use in source · pages.tex, line 100
ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Lon…
Exact mathematical passage
D_3=\FF_5[\delta]/(\delta^3)
\Delta_mUse in source · 2 passages
Search aliases: Δ_m Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 545
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\Delta_m=(\ZZ/p^m\ZZ)^r
\varepsilonUse in source · 20 passages
Search aliases: ε varepsilon
Use in source · jordan_packets.tex, line 5
node[font=sffamilysmallbfseries] at (1.6,2.35) packet at z=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.45) 1;node at (7.1,1.45) η;node at (8.7,1.45) η^2;node at (10.3,1.45) 0; draw[->] (5.78,1.45)--(6.82,1.45) node[midway,above,font=scriptsize] H+1; draw…
Exact mathematical passage
\varepsilon
\varepsilon^2Use in source · 7 passages
Search aliases: ε^2 varepsilon
Use in source · pages.tex, line 71
NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_1/I_1)whot_ O(R_2/I_2) ≅ A/J ≅ O[ε]/(ε^2).
Exact mathematical passage
(R_1/I_1)\whot_{\OO}(R_2/I_2) \cong A/J \cong \OO[\varepsilon]/(\varepsilon^2).
\epsilon_\DeltaUse in source · 1 passage
Search aliases: ϵ_Δ epsilon Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 478
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
\epsilon_\Delta q_*=\epsilon_{\Delta'}.
\epsilon_{\Delta'}Use in source · 1 passage
Search aliases: ϵ_Δ' epsilon Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 478
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
\epsilon_\Delta q_*=\epsilon_{\Delta'}.
\etaUse in source · 4 passages
Search aliases: η eta
Use in source · jordan_packets.tex, line 8
] at (1.6,2.35) packet at z=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.45) 1;node at (7.1,1.45) η;node at (8.7,1.45) η^2;node at (10.3,1.45) 0; draw[->] (5.78,1.45)--(6.82,1.45) node[midway,above,font=scriptsize] H+1; draw[->] (7.38,1.45)--(8.42,1.45) n…
Exact mathematical passage
\eta
\eta^2Use in source · 2 passages
Search aliases: η^2 eta
Use in source · jordan_packets.tex, line 8
=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.45) 1;node at (7.1,1.45) η;node at (8.7,1.45) η^2;node at (10.3,1.45) 0; draw[->] (5.78,1.45)--(6.82,1.45) node[midway,above,font=scriptsize] H+1; draw[->] (7.38,1.45)--(8.42,1.45) node[midway,above,font=scrip…
Exact mathematical passage
\eta^2
\eta^3Use in source · 1 passage
Search aliases: η^3 eta
Use in source · pages.tex, line 395
% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ underbrace Q[ε]/(ε^2)_T_+, z=1+ε × underbrace Q[η]/(η^3)_T_-, z=-1+η. Multiplication by z is an operator H. In the ordered local bases 1,ε and 1,η,η^2, it…
Exact mathematical passage
T\cong \underbrace{\QQ[\varepsilon]/(\varepsilon^2)}_{T_+,\ z=1+\varepsilon} \times \underbrace{\QQ[\eta]/(\eta^3)}_{T_-,\ z=-1+\eta}.
\Lambda^dUse in source · 1 passage
Search aliases: Λ^d Lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 541
Choose a Λ basis of M. Tensoring the resulting isomorphism M≅Λ^d along augmentation gives O^d. Tensoring by Λ/ a is the same as quotienting by a M.
Exact mathematical passage
M\cong\Lambda^d
\Lambda^rUse in source · 1 passage
Search aliases: Λ^r Lambda
Use in source · pages.tex, line 718
headRecover a free carrier, with its base ring named For Λ= Z_2[Δ] and the right module M=Λ^r, M⊗_Λ,ϵ Z_2 ≅ M/Mkerϵ≅ Z_2^r. The quotient applies augmentation to each basis coordinate. This remains valid for nonabelian finite two groups with the stated right-module convention. It establishes the rank over the coefficient ring after coinvariants, without establishing full support over a separate deformation ring.
Exact mathematical passage
M=\Lambda^r
\lambda_0Use in source · 4 passages
Search aliases: λ_0 lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1526
[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.
Exact mathematical passage
\lambda_0
\lambda_1Use in source · 1 passage
Search aliases: λ_1 lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1064
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
\frakm = (T_1-\lambda_1,\ldots,T_d-\lambda_d),
\lambda_dUse in source · 1 passage
Search aliases: λ_d lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1064
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
\frakm = (T_1-\lambda_1,\ldots,T_d-\lambda_d),
\lambda_iUse in source · 5 passages
Search aliases: λ_i lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1070
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
\boxed{ e_{\frakm}V = \left\{ v\in V: (T_i-\lambda_i)^Nv=0 \text{ for every }i \right\}. }
\muUse in source · 7 passages
Search aliases: μ mu
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 628
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\mu_{\frakp}Use in source · 2 passages
Search aliases: μ_frakp μ_ p mu frakp mu mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 628
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\mu_{\p}Use in source · 2 passages
Search aliases: μ_p mu p
Use in source · pages.tex, line 189
Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full algebraic loci over any field.
Exact mathematical passage
\mu_{\p}(M)=2-\rank A(\p).
\piUse in source · 22 passages
Search aliases: π pi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1349
tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X…
Exact mathematical passage
\pi_x
\varpiUse in source · 1 passage
Search aliases: ϖ varpi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 201
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
\varpi
\pi_xUse in source · 6 passages
Search aliases: π_x pi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1349
tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X…
Exact mathematical passage
\pi_x
\rho_1Use in source · 2 passages
Search aliases: ρ_1 rho
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1305
[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows from Chebotarev density and the Brauer Nesbitt theorem [source reference].
Exact mathematical passage
\rho_1,\rho_2:G_{F,S}\longrightarrow\GL_n(E')
\rho_2Use in source · 2 passages
Search aliases: ρ_2 rho
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1305
[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows from Chebotarev density and the Brauer Nesbitt theorem [source reference].
Exact mathematical passage
\rho_1,\rho_2:G_{F,S}\longrightarrow\GL_n(E')
\rho_x^{\mathrm{ss}}Use in source · 3 passages
Search aliases: ρ_x^ss rho mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1361
tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\boxed{ \rho_x^{\mathrm{ss}} \cong \rho_{\pi_x}. }
\rho_\lambdaUse in source · 9 passages
Search aliases: ρ_λ rho lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1492
[Conditional simultaneous companion completion] Under the coefficient and finite admissibility hypotheses above, one can choose continuous representations ρ_λ:Γ⟶GL_D( O_λ) (λ∈Λ) whose reduction class belongs to X_λ,n for every positive n. They retain the declared residual class. After applying i_λ, they are continuous, trivial on H_λ, and satisfy det (X- i_λ(ρ_λ(γ_v))) =e_λ(P_v), v∈Ω_λ. Their integral evaluation points factor through every actual quotient B_λ/J_λ,u.
Exact mathematical passage
\rho_\lambda:\Gamma\longrightarrow\GL_D(\OO_\lambda) \qquad(\lambda\in\Lambda)
\rho_{\lambda_0}Use in source · 1 passage
Search aliases: ρ_λ_0 rho lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1526
[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.
Exact mathematical passage
\rho_{\lambda_0}
\rho_{\pi,\lambda}Use in source · 1 passage
Search aliases: ρ_π,λ rho pi lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1747
For propagation, suppose both a semisimple compatible family {ρ_λ} containing ρ_x and a family {ρ_π,λ} attached to this same π already exist. Require the distinguished members to be ρ_x and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of [source reference], with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then ρ_λ≅ρ_π,λ^ss for every λ.
Exact mathematical passage
\{\rho_{\pi,\lambda}\}
\rho_{\pi,\lambda}^{\mathrm{ss}}Use in source · 3 passages
Search aliases: ρ_π,λ^ss rho pi lambda mathrm
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1527
be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.
Exact mathematical passage
\boxed{ \rho_\lambda \cong \rho_{\pi,\lambda}^{\mathrm{ss}}. }
\rho_{\pi_x}Use in source · 5 passages
Search aliases: ρ_π_x rho pi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1350
deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebot…
Exact mathematical passage
\rho_{\pi_x}:G_{F,S}\longrightarrow\GL_n(E')
\varphiUse in source · 5 passages
Search aliases: φ varphi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 921
[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).
Exact mathematical passage
\varphi:R\to\TT
\varphi^*Use in source · 2 passages
Search aliases: φ^* varphi
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 922
[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).
Exact mathematical passage
\varphi^*:\Spec\TT\longrightarrow\Spec R
\OmegaUse in source · 8 passages
Search aliases: Ω Omega
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1478
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
\Omega
\omegaUse in source · 5 passages
Search aliases: ω omega
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1546
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\omega_1,\ldots,\omega_r\in\Omega
\omega_1Use in source · 1 passage
Search aliases: ω_1 omega
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1546
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\omega_1,\ldots,\omega_r\in\Omega
\omega_jUse in source · 1 passage
Search aliases: ω_j omega
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1547
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\left(t_i(\omega_j)\right)_{j,i}
\omega_rUse in source · 1 passage
Search aliases: ω_r omega
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1546
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\omega_1,\ldots,\omega_r\in\Omega
\Omega_\lambdaUse in source · 3 passages
Search aliases: Ω_λ Omega lambda
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1478
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
\Omega_\lambda\subseteq\Omega
\ldotsUse in source · 18 passages
Search aliases: … ldots
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 257
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R_1,\ldots,R_s
\ellUse in source · 3 passages
Search aliases: ℓ ell
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1255
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
V=\ell^2(\NN)
\ell^2Use in source · 3 passages
Search aliases: ℓ^2 ell
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1255
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
V=\ell^2(\NN)
\twoheadrightarrowUse in source · 3 passages
Search aliases: ↠ twoheadrightarrow
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 967
[A quotient can erase a component] Let R=k[[x,y]]/(xy) and let T=R/(x)≅ k[[y]]. Viewed as an R module, T has support V(x), which misses the generic point (y) of the other irreducible component. Thus a surjection R↠ T does not by itself imply full component occupancy.
Exact mathematical passage
R\twoheadrightarrow\TT
\mapstoUse in source · 16 passages
Search aliases: ↦ mapsto
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 420
c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.
Exact mathematical passage
(z,(a,b))\mapsto zZ+aA+bB
\varnothingUse in source · 2 passages
Search aliases: ∅ varnothing
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1443
[Companion branch criterion] Suppose every level X_λ,m is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then varprojlim_mX_λ,m ≠∅.
Exact mathematical passage
\boxed{ \varprojlim_mX_{\lambda,m} \neq\varnothing. }
\prodUse in source · 3 passages
Search aliases: ∏ prod
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
\prod_{\frakm\in\Max\TT}Use in source · 1 passage
Search aliases: ∏_frakm∈MaxTT ∏_ m∈Max T prod frakm in Max TT prod mathfrak in operatorname mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1024
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
\prod_iUse in source · 1 passage
Search aliases: ∏_i prod
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 511
In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique maximal ideal.
Exact mathematical passage
\prod_ip^{n_i}=\abs{\Delta}
\prod_{i=1}^{r}Use in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 489
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
\sum_{\frakm}Use in source · 1 passage
Search aliases: ∑_frakm ∑_ m sum frakm sum mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1032
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
\sum_{g\in\Delta}Use in source · 1 passage
Search aliases: ∑_g∈Δ sum in Delta
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 465
Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.
Exact mathematical passage
\epsilon:\Lambda\longrightarrow\OO, \qquad \sum_{g\in\Delta}a_gg\longmapsto\sum_{g\in\Delta}a_g.
\sum_hUse in source · 1 passage
Search aliases: ∑_h sum
Use in source · pages.tex, line 693
headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.
Exact mathematical passage
u=-6^{-1}\sum_h f(h)
\sum_{h\in S_3}Use in source · 1 passage
Search aliases: ∑_h∈ S_3 sum in
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 404
A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomology vanishings.
Exact mathematical passage
u=-6^{-1}\sum_{h\in S_3}f(h).
\sum_{i,j\ge0}Use in source · 1 passage
Search aliases: ∑_i,j≥0 sum ge
Use in source · pages.tex, line 81
headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.
Exact mathematical passage
f=\sum_{i,j\ge0}a_{ij}x^iy^j
\sum_{j\ge0}Use in source · 2 passages
Search aliases: ∑_j≥0 sum ge
Use in source · pages.tex, line 142
headTwo explicit 5-adic numbers Choose the canonical digits α_j=(j^2+1)bmod5, β_j=(2j+1)bmod5, with values in {0,1,2,3,4}. Set a=∑_j≥0α_j5^j and b=∑_j≥0β_j5^j. The element a+ε b is determined by the following coefficient array.
Exact mathematical passage
a=\sum_{j\ge0}\alpha_j5^j
\setminusUse in source · 2 passages
Search aliases: ∖ setminus
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 596
Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.
Exact mathematical passage
a\in\Ann_R(M)\setminus\frakp
\capUse in source · 4 passages
Search aliases: ∩ cap
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 706
[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).
Exact mathematical passage
\boxed{ \Supp_{R/I}(M/IM) = \Supp_R(M)\cap V(I). }
\cupUse in source · 3 passages
Search aliases: ∪ cup
Use in source · pages.tex, line 205
bottomrule Thus Supp(M)=V(xy)=V(x)∪ V(y), whereas the locus where the fiber needs at least two generators is V(x,y). The latter is a higher Fitting stratum. For r>2, the locus μ_p(M)≥ r is empty, corresponding to the unit Fitting ideal.
Exact mathematical passage
\Supp(M)=V(xy)=V(x)\cup V(y)
\simUse in source · 1 passage
Search aliases: ∼ sim
Use in source · pages.tex, line 283
headAn isomorphism changes coordinates, not occupancy Over a field of characteristic other than two, the isomorphism k[x,y]xrightarrow∼k[a,b], x↦ a+b, y↦ a-b sends the Fitting ideals to (a^2-b^2) and (a,b). The two axes become the two lines a=b and a=-b. No previously missing module vector is created by this relabeling.
Exact mathematical passage
k[x,y]\xrightarrow{\sim}k[a,b],\qquad x\mapsto a+b,\quad y\mapsto a-b
\equivUse in source · 3 passages
Search aliases: ≡ equiv
Use in source · pages.tex, line 416
headThe projectors can be written explicitly Modulo (z-1)^2(z+1)^3, put e_+(z)=frac(5-3z)(z+1)^316, e_-(z)=1-e_+(z). Then e_+≡1pmod(z-1)^2 and e_+≡0pmod(z+1)^3. Consequently e_+(H)=diag(1,1,0,0,0), e_-(H)=diag(0,0,1,1,1). The packet dimensions are two and three. The ordinary eigenspaces each have dimension one; the generalized eigenspaces have dimensions two and three.
Exact mathematical passage
e_+\equiv1\pmod{(z-1)^2}
\leqUse in source · 2 passages
Search aliases: ≤ leq
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 648
Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equivalent to the vanishing of every minor of size n-r+1. Those minors generate Fitt_r-1^R(M).
Exact mathematical passage
1\leq r\leq n
\geUse in source · 13 passages
Search aliases: ≥ ge
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 436
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
n\ge1
\supseteqUse in source · 1 passage
Search aliases: ⊇ supseteq
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 716
Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.
Exact mathematical passage
\frakp\supseteq I
\bigoplusUse in source · 1 passage
Search aliases: ⊕ bigoplus
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1043
[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).
Exact mathematical passage
\boxed{ V = \bigoplus_{\frakm\in\Max\TT}e_{\frakm}V. }
\oplusUse in source · 5 passages
Search aliases: ⊕ oplus
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 386
[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.
Exact mathematical passage
\ad^0V\cong\operatorname{sgn}\oplus V.
\bigoplus_{\frakm\in\Max\TT}Use in source · 1 passage
Search aliases: ⊕_frakm∈MaxTT ⊕_ m∈Max T bigoplus frakm in Max TT bigoplus mathfrak in operatorname mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1043
[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).
Exact mathematical passage
\boxed{ V = \bigoplus_{\frakm\in\Max\TT}e_{\frakm}V. }
\otimes_AUse in source · 1 passage
Search aliases: ⊗_A otimes
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 312
Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/o…
Exact mathematical passage
B\otimes_A(A/J)\cong B/JB
\bigotimes_{i=1}^sUse in source · 1 passage
Search aliases: ⊗_i=1^s bigotimes
Use in source · pages.tex, line 649
-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal of definition: S=⊗_i=1^s R_i, a=m_ OS+∑_im_R_iS, L=∑_i I_iS. The comparison uses the a-adic additive uniformity on S, the image ideal topology on S/L…
Exact mathematical passage
S=\bigotimes_{i=1}^s R_i,\qquad \mathfrak a=\m_{\OO}S+\sum_i\m_{R_i}S,\qquad L=\sum_i I_iS.
\otimes_RUse in source · 7 passages
Search aliases: ⊗_R otimes
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 628
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\otimes_{R,x}Use in source · 1 passage
Search aliases: ⊗_R,x otimes
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 977
[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).
Exact mathematical passage
\boxed{ M\otimes_{R,x}E'\neq0 \quad\Longleftrightarrow\quad \frakp\in\Supp_R(M). }
\otimes_{R/I}Use in source · 1 passage
Search aliases: ⊗_R/I otimes
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 852
Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fi…
Exact mathematical passage
\kappa(q)\otimes_{R/I}\bigl((R/I)\otimes_RP\bigr) \cong\kappa(q)\otimes_RP \cong\kappa(q)\otimes_{\kappa(\frakp)} \bigl(\kappa(\frakp)\otimes_RP\bigr).
\otimes_R^{\mathbf L}Use in source · 2 passages
Search aliases: ⊗_R^ L otimes mathbf
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 849
Choose a bounded complex P of finite projective R modules representing C. Scalar extension to a commutative R algebra S is computed degree by degree as S⊗_RP; the canonical symmetry of tensor products gives the displayed order C⊗_R^ LS. Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism between two such projective models lifts to a homotopy equivalence, which scalar extension preserves. The resulting derived object and its residue field…
Exact mathematical passage
C\otimes_R^{\mathbf L}S
\otimes_{T,\,z\mapsto1}Use in source · 1 passage
Search aliases: ⊗_T, z↦1 otimes mapsto
Use in source · pages.tex, line 460
headA generalized packet is not its residue-field fiber At the point z=1, the full local packet e_+V has dimension two, but its scalar fiber is V⊗_T, z↦1 Q ≅ T_+/(ε)≅ Q. The packet has the basis 1,ε. Its scalar fiber kills ε. The ordinary eigenspace is instead the kernel of multiplication by ε, spanned by ε. In this example the last two spaces have the same dimension, but they are obtained by different operations.
Exact mathematical passage
V\otimes_{T,\,z\mapsto1}\QQ \cong T_+/(\varepsilon)\cong\QQ.
\otimes_{\TT,x}Use in source · 3 passages
Search aliases: ⊗_TT,x ⊗_ T,x otimes TT otimes mathbb
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 999
[Supported deformation points give nonzero Hecke fibers] Let R≅ T, and let M be a finite Hecke module. If a point x:R→ E' lies in Supp_R(M), then the corresponding Hecke fiber M⊗_ T,xE' is nonzero.
Exact mathematical passage
M\otimes_{\TT,x}E'
\otimes_{\kappa(\frakp)}Use in source · 1 passage
Search aliases: ⊗_κ(frakp) ⊗_κ( p) otimes kappa frakp otimes kappa mathfrak
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 852
Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left hand complex is thus nonzero in the derived category exactly when the residue field fi…
Exact mathematical passage
\kappa(q)\otimes_{R/I}\bigl((R/I)\otimes_RP\bigr) \cong\kappa(q)\otimes_RP \cong\kappa(q)\otimes_{\kappa(\frakp)} \bigl(\kappa(\frakp)\otimes_RP\bigr).
\bigcap_{m\ge1}Use in source · 1 passage
Search aliases: ⋂_m≥1 bigcap ge
Use in source · pages.tex, line 584
headWhy every infinite tail can be nonzero while their intersection vanishes Let P_m project onto the closed span of e_m,e_m+1,…. Each range is nonzero. But a vector belonging to every range has its j-th coordinate zero once m>j. Every coordinate is therefore zero: ⋂_m≥1P_mℓ^2( N)={0}.
Exact mathematical passage
\bigcap_{m\ge1}P_m\ell^2(\mathbb N)=\{0\}.
\bigcap_{m\geq1}Use in source · 3 passages
Search aliases: ⋂_m≥1 bigcap geq
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1154
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
\boxed{ \bigcap_{m\geq1}e_mV \neq0. }
\bigcap_{n\ge m}Use in source · 1 passage
Search aliases: ⋂_n≥ m bigcap ge
Use in source · pages.tex, line 549
headWhat the stable core measures In a general inverse system of finite-dimensional spaces, Y_m=⋂_n≥ mim(f_n,m) contains exactly the vectors visible from arbitrarily remote levels. The paper proves that the transition maps restrict to surjections Y_m+1↠ Y_m. A prescribed nonzero vector in Y_1 can then be lifted recursively to a coherent nonzero sequence.
Exact mathematical passage
Y_m=\bigcap_{n\ge m}\im(f_{n,m})
\bigcap_{n\ge1}Use in source · 1 passage
Search aliases: ⋂_n≥1 bigcap ge
Use in source · pages.tex, line 788
headThe reconstruction argument is coefficientwise Let A_n be compatible matrices modulo 5^n, each satisfying these four equations. Completeness reconstructs A∈ M_2( Z_5). For each polynomial f∈ J, evaluation commutes with reduction, so f(A)∈⋂_n≥15^n Z_5={0}. Separatedness supplies the last equality. Thus the actual evaluation map B→ Z_5 kills J and factors uniquely through B/J. If inverse matrices were reconstructed as well, this is an actual representation in GL_2( Z_5).
Exact mathematical passage
f(A)\in\bigcap_{n\ge1}5^n\ZZ_5=\{0\}.
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 825
[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].
Exact mathematical passage
\Supp_R(C) = \bigcup_i\Supp_R H^i(C).
\bigcup_iUse in source · 1 passage
Search aliases: ⋃_i bigcup
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 825
[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].
Exact mathematical passage
\Supp_R(C) = \bigcup_i\Supp_R H^i(C).
\langleUse in source · 2 passages
Search aliases: ⟨ langle
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1579
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
C_2=\langle g\rangle
\rangleUse in source · 2 passages
Search aliases: ⟩ rangle
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 1579
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
C_2=\langle g\rangle
\longmapstoUse in source · 6 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · Paper_XII_Traditional_Final.tex, line 394
The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles ar…
Exact mathematical passage
(a,b)\longmapsto(a,-a+b,-b)
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- papers/paper-xii/current/core/publication/source/visual_guide/graphics/core_nilpotent.tex · 17 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/core_projection.tex · 16 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/derived_fibers.tex · 8 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/diagonal_slice.tex · 4 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/finite_forest.tex · 12 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/frames_example.tex · 6 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/infinite_tail.tex · 20 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/jordan_packets.tex · 14 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/monomial_quotient.tex · 28 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/nested_finite.tex · 11 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/node_spectrum.tex · 8 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/patching_branches.tex · 5 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/points_curvatures.tex · 22 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/points_slopes.tex · 26 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/polynomial_effectivity.tex · 6 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/support_axes.tex · 7 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/graphics/trace_separator.tex · 6 mathematical passages
- papers/paper-xii/current/core/publication/source/visual_guide/source/pages.tex · 468 mathematical passages
PDF reading editions (21)
The text index does not extract PDF glyphs. Check the reading edition for notation in figures, tables, and PDF-only material.
- Paper_XII_Traditional_v1.0.0.pdf
- Paper_XII_Visual_Guide_v1.0.0.pdf
- Paper_XII_Lean_Guide_v1.0.0.pdf
- 01_global_gates.pdf
- 18_claim_boundary.pdf
- 02_local_quotient_packets.pdf
- 03_adequacy_packets.pdf
- 04_auxiliary_group_tower.pdf
- 05_augmentation_algebra.pdf
- 06_patching_support_descent.pdf
- 07_freeness_counterexample.pdf
- 08_rt_support_transport.pdf
- 09_hecke_artin_packets.pdf
- 10_generalized_eigenspaces.pdf
- 11_finite_occupancy_staircase.pdf
- 12_stable_core_inverse_limit.pdf
- 13_tail_escape.pdf
- 16_pointwise_automorphy.pdf
- 14_companion_spine.pdf
- 15_coefficient_two_boundary.pdf
- 17_final_completion.pdf