block12_section.tex
Stratified Universality
block12_section.tex
This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.
Included manuscript sections
1 \section{Finite Galois, profinite, and Tannakian resolved symmetry}
2 \label{sec:block12-galois-tannakian}
3
4 Galois-Tannakian atlas develops the final declared stratum in the present eight-stratum program: finite semisimple Galois charts, cohomological descent, profinite refinement, and the tensor-fiber boundary at which a symmetry group can be reconstructed. The strongest Galois source already contains Reynolds projectors, the subgroup--fixed-field correspondence, normality and quotient compression, conjugate projector atlases, Mackey double-coset packets, ramification details, Haar projectors, Hilbert 90, the Brauer obstruction, finite \(\acute{e}\)tale permutation resolutions, and neutral Tannakian reconstruction \cite{GAL}. The purpose here is to place those results into the same passport, bridge, obstruction, and effective-descent language used by the preceding sections.
5
6 The scope is deliberately semisimple or tensor-exact. For finite group averaging, the coefficient field has characteristic zero or characteristic prime to the relevant group order. Linear semilinear descent is taken over a finite Galois extension. Projective descent is typed separately from vector descent. Tannakian reconstruction is asserted only after a faithful exact tensor fiber functor has been supplied. Modular fixed points, wild integral descent, nonneutral fiber functors, and derived or flat replacements for failed averaging remain outside this block.
7
8 \subsection{Finite subgroup, fixed-field, and Reynolds-projector charts}
9 \label{subsec:block12-reynolds}
10
11 Let \(L/K\) be a finite Galois extension with group
12 \[
13 G=\operatorname{Gal}(L/K).
14 \]
15 Let \(V\) be a finite-dimensional representation of \(G\) over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup \(H\leq G\), define
16 \[
17 \boxed{
18 P_H=\frac1{|H|}\sum_{h\in H}U_h.}
19 \tag{12.1}
20 \]
21
22 \begin{theorem}[Reynolds fixed packet and reverse order]
23 \label{thm:block12-reynolds}
24 The operator \(P_H\) is the projector onto the fixed packet
25 \[
26 V^H=\{v:U_hv=v\text{ for all }h\in H\}.
27 \]
28 If \(H\leq J\), then
29 \[
30 \boxed{P_HP_J=P_JP_H=P_J.}
31 \tag{12.2}
32 \]
33 Consequently, subgroup inclusion reverses the fixed-projector order.
34 \end{theorem}
35
36 \begin{proof}
37 The average is invariant under left and right multiplication by elements of \(H\). Therefore \(P_H^2=P_H\), and its range is exactly the fixed subspace. If \(H\leq J\), then \(V^J\subseteq V^H\). The two projections are nested and multiply to the projector with smaller range.
38 \end{proof}
39
40 In the faithful regular realization, or in the field realization supplied by a normal basis, the projector determines the subgroup. Indeed, the support of the group-algebra idempotent
41 \[
42 e_H=\frac1{|H|}\sum_{h\in H}h
43 \]
44 is exactly \(H\).
45
46 \begin{theorem}[Normalized finite Galois chart]
47 \label{thm:block12-normalized-galois-chart}
48 Fix the finite Galois extension \(L/K\). The following data determine one another exactly:
49 \[
50 \boxed{
51 H
52 \quad\longleftrightarrow\quad
53 L^H
54 \quad\longleftrightarrow\quad
55 P_H
56 }
57 \tag{12.3}
58 \]
59 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing:
60 \[
61 H\leq J
62 \quad\Longleftrightarrow\quad
63 L^H\supseteq L^J
64 \quad\Longleftrightarrow\quad
65 P_H\geq P_J.
66 \]
67 Thus the subgroup poset, the intermediate-field poset, and the faithful fixed-projector poset form three normalized charts of one finite semisimple Galois object.
68 \end{theorem}
69
70 \begin{proof}
71 The finite Galois correspondence gives \(H=\operatorname{Gal}(L/L^H)\) and \(E=L^{\operatorname{Gal}(L/E)}\). The range of \(P_H\) is \(L^H\) after faithful scalar extension, and the support of \(e_H\) recovers \(H\). All three assignments are therefore mutually inverse on the declared charts.
72 \end{proof}
73
74 \begin{figure}[htbp]
75 \centering
76 \includegraphics[width=.97\textwidth]{galois_subgroup_atlas.pdf}
77 \caption{The finite subgroup, Reynolds-projector, and fixed-field charts are reversible in a faithful semisimple realization. Normality is the additional condition required for quotient compression.}
78 \label{fig:block12-subgroup-atlas}
79 \end{figure}
80
81 \subsection{Compatibility, normality, and conjugate atlases}
82 \label{subsec:block12-normality}
83
84 For two subgroups \(H,J\leq G\), the common fixed packet is always
85 \[
86 V^H\cap V^J=V^{\langle H,J\rangle}.
87 \]
88 The two averaging projectors need not commute.
89
90 \begin{theorem}[Permutable subgroups and exact common refinement]
91 \label{thm:block12-permutable}
92 In a faithful regular realization, the following are equivalent:
93 \begin{enumerate}
94 \item \(HJ=JH\);
95 \item \(HJ\) is a subgroup of \(G\);
96 \item \(P_HP_J=P_JP_H\).
97 \end{enumerate}
98 When these conditions hold,
99 \[
100 \boxed{P_HP_J=P_{HJ}.}
101 \tag{12.4}
102 \]
103 If they fail, \(P_HP_J\) and \(P_JP_H\) are mixed atlas corners rather than projectors.
104 \end{theorem}
105
106 \begin{proof}
107 If \(HJ=JH\), every element of \(HJ\) has \(|H\cap J|\) representations as \(hj\). Multiplying the two group averages therefore gives the average over \(HJ\). Conversely, the support of \(e_He_J\) is \(HJ\), while the support of \(e_Je_H\) is \(JH\). Equality in the regular group algebra forces equality of the product sets.
108 \end{proof}
109
110 \begin{theorem}[Normality and quotient compression]
111 \label{thm:block12-normality}
112 For a subgroup \(H\leq G\), the following conditions are equivalent:
113 \begin{enumerate}
114 \item \(H\triangleleft G\);
115 \item \(e_H\) is central in the group algebra;
116 \item \(P_HU_g=U_gP_H\) for every \(g\in G\);
117 \item the fixed packet \(V^H\) is \(G\)-invariant;
118 \item the fixed field \(L^H\) is Galois over \(K\).
119 \end{enumerate}
120 When they hold, the compressed action on \(V^H\) factors through \(G/H\), and
121 \[
122 \boxed{\operatorname{Gal}(L^H/K)\cong G/H.}
123 \tag{12.5}
124 \]
125 \end{theorem}
126
127 \begin{proof}
128 Conjugation sends \(e_H\) to \(e_{gHg^{-1}}\). Centrality is therefore equivalent to normality. The represented identity
129 \[
130 U_gP_HU_g^{-1}=P_{gHg^{-1}}
131 \]
132 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.
133 \end{proof}
134
135 If \(H\) is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put
136 \[
137 \operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1},
138 \qquad
139 \operatorname{Ncl}_G(H)=\left\langle gHg^{-1}:g\in G\right\rangle.
140 \]
141
142 \begin{theorem}[Canonical envelopes of a nonnormal Galois atlas]
143 \label{thm:block12-core-normal-closure}
144 Let \(M=L^H\). Then
145 \[
146 \boxed{
147 L^{\operatorname{Core}_G(H)}
148 =\prod_{g\in G}g(M)}
149 \tag{12.6}
150 \]
151 is the smallest Galois extension inside \(L\) containing every conjugate chart, while
152 \[
153 \boxed{
154 L^{\operatorname{Ncl}_G(H)}
155 =\bigcap_{g\in G}g(M)}
156 \tag{12.7}
157 \]
158 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of the conjugate atlas.
159 \end{theorem}
160
161 \begin{proof}
162 Composita of intermediate fields correspond to intersections of subgroups, while intersections of fields correspond to generated subgroups. The core is the largest normal subgroup inside \(H\), and the normal closure is the smallest normal subgroup containing \(H\).
163 \end{proof}
164
165 \begin{example}[The \(S_3\) nonnormal atlas]
166 Let \(G=S_3\) and let \(H\) be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of \(S_3\). By contrast, the order-three subgroup \(A_3\) is normal and its fixed chart carries the quotient symmetry \(S_3/A_3\cong C_2\).
167 \end{example}
168
169 \subsection{Double cosets as exact transition packets}
170 \label{subsec:block12-mackey}
171
172 Restriction and induction connect subgroup charts that need not commute. Their exact address set is the double-coset space
173 \[
174 K\backslash G/H.
175 \]
176
177 \begin{theorem}[Mackey transition-packet decomposition]
178 \label{thm:block12-mackey}
179 Let \(H,K\leq G\), let \(W\) be a finite-dimensional representation of \(H\), and choose representatives \(x\) for \(K\backslash G/H\). Then
180 \[
181 \boxed{
182 \operatorname{Res}_K^G\operatorname{Ind}_H^GW
183 \cong
184 \bigoplus_{x\in K\backslash G/H}
185 \operatorname{Ind}_{K\cap xHx^{-1}}^K
186 \left({}^x\operatorname{Res}_{H\cap x^{-1}Kx}^HW\right).}
187 \tag{12.8}
188 \]
189 Each double coset is one exact transfer packet between the \(H\)- and \(K\)-charts.
190 \end{theorem}
191
192 \begin{proof}
193 The induced representation is a function space on \(G\) with \(H\)-covariance. Functions supported on different double cosets are invariant direct summands under \(K\). Identifying the summand on \(KxH\) with an induced representation from \(K\cap xHx^{-1}\) gives the decomposition.
194 \end{proof}
195
196 The dimension identity for the trivial \(H\)-representation is
197 \[
198 \boxed{
199 [G:H]
200 =
201 \sum_{x\in K\backslash G/H}
202 [K:K\cap xHx^{-1}].}
203 \tag{12.9}
204 \]
205 This is the finite packet-count law audited by the Galois-Tannakian atlas verifier.
206
207 \begin{figure}[htbp]
208 \centering
209 \includegraphics[width=.97\textwidth]{galois_mackey_packets.pdf}
210 \caption{A nonnormal fixed object is an atlas of conjugate charts. Core and normal closure give two global envelopes, while double cosets give exact mixed transition packets.}
211 \label{fig:block12-mackey}
212 \end{figure}
213
214 \input{generated/block12_group_table.tex}
215
216 \subsection{Ramification as a nested resolution}
217 \label{subsec:block12-ramification}
218
219 Let \(L/K\) be a finite Galois extension of local fields. Write
220 \[
221 G_0\supseteq G_1\supseteq G_2\supseteq\cdots
222 \]
223 for the lower ramification filtration after inertia, and let \(V\) be a finite-dimensional complex representation of \(G\). Define
224 \[
225 P_i=\frac1{|G_i|}\sum_{g\in G_i}U_g.
226 \]
227 The ramification groups are nested and normal, so
228 \[
229 P_0\leq P_1\leq P_2\leq\cdots\leq I.
230 \]
231 Put
232 \[
233 D_i=P_{i+1}-P_i.
234 \]
235
236 \begin{theorem}[Orthogonal ramification details]
237 \label{thm:block12-ramification-details}
238 The operators \(D_i\) are mutually orthogonal projectors, and
239 \[
240 \boxed{
241 V=V^{G_0}\oplus\bigoplus_i D_iV.}
242 \tag{12.10}
243 \]
244 Every detail packet is stable under the quotient action permitted by the corresponding normal subgroup.
245 \end{theorem}
246
247 \begin{proof}
248 Nested orthogonal projectors commute, and the difference of two nested projectors is a projector onto the orthogonal difference of their ranges. The decomposition follows by telescoping.
249 \end{proof}
250
251 \begin{theorem}[Artin conductor as weighted unresolved rank]
252 \label{thm:block12-artin-conductor}
253 The Artin and Swan conductors satisfy
254 \[
255 \boxed{
256 a(V)=\sum_{i\geq0}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i),}
257 \tag{12.11}
258 \]
259 \[
260 \boxed{
261 \operatorname{Sw}(V)=\sum_{i\geq1}\frac{|G_i|}{|G_0|}\operatorname{Tr}(I-P_i).}
262 \tag{12.12}
263 \]
264 If \(d_j=\operatorname{rank}D_j\), then
265 \[
266 \boxed{
267 a(V)=\sum_jd_j\sum_{i=0}^{j}\frac{|G_i|}{|G_0|}.}
268 \tag{12.13}
269 \]
270 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.
271 \end{theorem}
272
273 \begin{proof}
274 The classical conductor formula uses \(\operatorname{codim}V^{G_i}\). Since \(P_i\) projects onto \(V^{G_i}\), this codimension is \(\operatorname{Tr}(I-P_i)\). Also
275 \[
276 \operatorname{codim}V^{G_i}=\sum_{j\geq i}d_j.
277 \]
278 Reversing the finite sums gives the detail formula.
279 \end{proof}
280
281 \begin{figure}[htbp]
282 \centering
283 \includegraphics[width=.92\textwidth]{galois_ramification_resolution.pdf}
284 \caption{Ramification subgroups give nested fixed projectors. Their orthogonal differences are the resolved tame and wild detail packets, and the conductor is their weighted unresolved rank.}
285 \label{fig:block12-ramification}
286 \end{figure}
287
288 \subsection{Profinite completion and finite quotient effectivity}
289 \label{subsec:block12-profinite}
290
291 Let \(G\) be a compact group and let \(U:G\to\mathcal U(\mathcal H)\) be strongly continuous. For a closed subgroup \(H\), normalized Haar measure defines
292 \[
293 \boxed{
294 P_H=\int_HU_h\,d\mu_H(h).}
295 \tag{12.14}
296 \]
297 The same averaging proof as in the finite case shows that \(P_H\) is the orthogonal projector onto \(\mathcal H^H\).
298
299 \begin{theorem}[Shrinking-subgroup reconstruction]
300 \label{thm:block12-shrinking-subgroups}
301 Let \(G\) be profinite and let
302 \[
303 U_1\supseteq U_2\supseteq\cdots
304 \]
305 be a neighborhood basis of the identity consisting of open normal subgroups. Then
306 \[
307 \boxed{P_{U_n}\longrightarrow I}
308 \tag{12.15}
309 \]
310 strongly in every strongly continuous unitary representation. More generally, if \(N\triangleleft G\) is closed and \(U_n/N\) shrinks to the identity in \(G/N\), then \(P_{U_n}\to P_N\) strongly.
311 \end{theorem}
312
313 \begin{proof}
314 For each \(v\in\mathcal H\), strong continuity gives
315 \[
316 \sup_{u\in U_n}\|U_uv-v\|\longrightarrow0.
317 \]
318 Averaging this estimate over \(U_n\) proves the first statement. Apply it to the quotient action on the \(N\)-fixed packet for the second.
319 \end{proof}
320
321 \begin{corollary}[Finite-dimensional representations occur at finite level]
322 \label{cor:block12-finite-level}
323 Every continuous finite-dimensional complex representation of a profinite group has finite image and factors through a finite quotient.
324 \end{corollary}
325
326 \begin{proof}
327 The image is a compact totally disconnected subgroup of \(\operatorname{GL}_n(\mathbb C)\). After conjugation it lies in a compact Lie group. A compact totally disconnected Lie group is finite.
328 \end{proof}
329
330 This finite-level factorization is a strong effectivity statement for finite-dimensional complex representations. It is not a statement that every integral, modular, or derived representation becomes semisimple at finite level.
331
332 \subsection{Semilinear vector descent and Hilbert 90}
333 \label{subsec:block12-hilbert90}
334
335 Let \(L/K\) be finite Galois with group \(G\). A semilinear \(G\)-object is an \(L\)-vector space \(V\) with maps
336 \[
337 \rho_\sigma:V\to V,
338 \qquad
339 \rho_\sigma(\lambda v)=\sigma(\lambda)\rho_\sigma(v),
340 \]
341 satisfying the group law. Scalar extension and fixed vectors define functors
342 \[
343 L\otimes_K-:\operatorname{Vect}_K\longrightarrow\operatorname{Semilin}_G(L),
344 \]
345 \[
346 (-)^G:\operatorname{Semilin}_G(L)\longrightarrow\operatorname{Vect}_K.
347 \]
348
349 \begin{theorem}[Effective finite Galois vector descent]
350 \label{thm:block12-vector-descent}
351 The two functors above are quasi-inverse equivalences:
352 \[
353 \boxed{
354 \operatorname{Vect}_K
355 \simeq
356 \operatorname{Semilin}_G(L).}
357 \tag{12.16}
358 \]
359 For every semilinear object \(V\), the fixed packet \(W=V^G\) satisfies
360 \[
361 \boxed{W\otimes_KL\cong V.}
362 \tag{12.17}
363 \]
364 The equivalence is effective on both objects and morphisms.
365 \end{theorem}
366
367 \begin{proof}
368 This is finite Galois descent. After an \(L\)-basis is chosen, the semilinear action is represented by a cocycle
369 \[
370 a_{\sigma\tau}=a_\sigma\sigma(a_\tau),
371 \qquad
372 a_\sigma\in\operatorname{GL}_n(L).
373 \]
374 Nonabelian Hilbert 90 gives \(B\in\operatorname{GL}_n(L)\) with
375 \[
376 a_\sigma=B^{-1}\sigma(B).
377 \]
378 The columns of \(B^{-1}\) applied to a \(K\)-basis are fixed by the twisted action. They form an \(n\)-dimensional fixed packet, and scalar extension reconstructs \(V\). An equivariant \(L\)-linear map restricts to a \(K\)-linear map on fixed packets, and scalar extension recovers it.
379 \end{proof}
380
381 A useful constructive form is obtained by choosing any matrix \(C\in M_n(L)\) and setting
382 \[
383 T(C)=\sum_{\sigma\in G}a_\sigma\sigma(C).
384 \]
385 Then
386 \[
387 a_\tau\tau(T(C))=T(C).
388 \]
389 Whenever \(T(C)\) is invertible,
390 \[
391 B=T(C)^{-1}
392 \]
393 is a Hilbert 90 trivialization. Standard descent guarantees that such a \(C\) exists.
394
395 \begin{definition}[Framed semilinear descent chart]
396 A framed descent chart retains an ordered \(K\)-basis of the fixed packet \(W\). Its scalar extension gives a distinguished \(L\)-basis of \(V\), and hence a literal cocycle matrix family.
397 \end{definition}
398
399 \begin{proposition}[Gauge and intrinsic automorphisms remain distinct]
400 \label{prop:block12-framed-unframed}
401 For fixed rank \(n\), nonabelian Hilbert 90 implies that every \(\operatorname{GL}_n(L)\)-cocycle is gauge equivalent to the trivial cocycle. The unframed descent groupoid is not contractible: its automorphism group is \(\operatorname{GL}_n(K)\). After a global \(K\)-frame is retained, the normalized fiber becomes contractible.
402 \end{proposition}
403
404 \begin{proof}
405 Hilbert 90 gives one isomorphism class. Automorphisms of the descended \(K\)-space are exactly \(\operatorname{GL}_n(K)\). A chosen ordered basis kills those automorphisms.
406 \end{proof}
407
408 \subsection{Projective descent and the Brauer obstruction}
409 \label{subsec:block12-brauer}
410
411 A projective cocycle takes values in \(\operatorname{PGL}_n(L)\). Choose lifts
412 \[
413 A_\sigma\in\operatorname{GL}_n(L).
414 \]
415 Their cocycle failure is scalar:
416 \[
417 \boxed{
418 c(\sigma,\tau)
419 =A_\sigma\sigma(A_\tau)A_{\sigma\tau}^{-1}
420 \in L^\times.}
421 \tag{12.18}
422 \]
423 Changing the lifts alters \(c\) by a scalar coboundary.
424
425 \begin{theorem}[Exact projective descent obstruction]
426 \label{thm:block12-brauer}
427 The scalar class
428 \[
429 \boxed{
430 [c]=\delta([\alpha])
431 \in H^2(G,L^\times)
432 =\operatorname{Br}(L/K)}
433 \tag{12.19}
434 \]
435 is independent of the chosen lifts. It vanishes exactly when the projective cocycle lifts to a linear semilinear cocycle. In that case \cref{thm:block12-vector-descent} produces a descended vector space. If it does not vanish, the descended object is a module over a central simple \(K\)-algebra split by \(L\), not an ordinary global vector packet.
436 \end{theorem}
437
438 \begin{proof}
439 Apply the connecting map in nonabelian cohomology to
440 \[
441 1\longrightarrow L^\times
442 \longrightarrow\operatorname{GL}_n(L)
443 \longrightarrow\operatorname{PGL}_n(L)
444 \longrightarrow1.
445 \]
446 Vanishing is equivalent to rescaling the lifts so that the scalar defect is removed. The lifted cocycle then descends by Hilbert 90. The nonzero classes are the relative Brauer classes of central simple algebras split by \(L\).
447 \end{proof}
448
449 \begin{example}[The quaternionic obstruction over \(\mathbb C/\mathbb R\)]
450 Let \(G=\{1,\sigma\}\), where \(\sigma\) is complex conjugation, and put
451 \[
452 A_\sigma=
453 \begin{pmatrix}
454 0&1\\
455 -1&0
456 \end{pmatrix}.
457 \]
458 Then
459 \[
460 A_\sigma\sigma(A_\sigma)=-I.
461 \]
462 The projective cocycle condition holds because \(-I\) is scalar. Its boundary is represented by \(-1\) in
463 \[
464 \mathbb R^\times/N_{\mathbb C/\mathbb R}(\mathbb C^\times).
465 \]
466 Every complex norm is positive, so \(-1\) is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing \(A_\sigma\) by
467 \[
468 \begin{pmatrix}0&1\\1&0\end{pmatrix}
469 \]
470 gives scalar defect \(+1\) and hence split vector descent.
471 \end{example}
472
473 \begin{figure}[htbp]
474 \centering
475 \includegraphics[width=.97\textwidth]{galois_descent_obstruction.pdf}
476 \caption{Linear cocycles are removed by Hilbert 90. Projective cocycles can satisfy every visible \(\operatorname{PGL}\) overlap law while retaining a nonzero degree-two Brauer obstruction.}
477 \label{fig:block12-descent-obstruction}
478 \end{figure}
479
480 \subsection{Finite \texorpdfstring{\(\acute{e}\)}{e}tale and neutral Tannakian symmetry}
481 \label{subsec:block12-tannakian}
482
483 For a connected scheme \(X\) with geometric point \(\bar x\), the fiber functor on finite \(\acute{e}\)tale covers identifies that category with finite continuous \(\pi_1^{\acute{e}t}(X,\bar x)\)-sets. Every finite fiber \(S\) produces a permutation resolution on \(\ell^2(S)\): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms give equivariant packet transfers. The profinite fundamental group is the automorphism group of this fiber functor.
484
485 The tensor-linear version requires more structure.
486
487 \begin{theorem}[Neutral Tannakian reconstruction]
488 \label{thm:block12-tannakian}
489 Let \(\mathcal C\) be a neutral Tannakian category over \(k\), and let
490 \[
491 \omega:\mathcal C\longrightarrow\operatorname{Vect}_k
492 \]
493 be a faithful exact tensor fiber functor. Then
494 \[
495 \boxed{
496 \mathcal G=\operatorname{Aut}^{\otimes}(\omega)}
497 \tag{12.20}
498 \]
499 is an affine group scheme, and
500 \[
501 \boxed{
502 \mathcal C\simeq\operatorname{Rep}_k(\mathcal G)}
503 \tag{12.21}
504 \]
505 as tensor categories. Declared packet idempotents and refinement maps preserved by \(\omega\) become tensor-compatible concrete projector resolutions in every fiber.
506 \end{theorem}
507
508 \begin{proof}
509 The group scheme and tensor equivalence are the neutral Tannakian reconstruction theorem. Additivity and exact tensor preservation transport the declared idempotent and refinement equations to the fibers.
510 \end{proof}
511
512 \begin{boundary}[Faithful corner realization is not a fiber functor]
513 A faithful additive or Hilbert-space realization can preserve packet idempotents, transfer blocks, and operator equations without preserving tensor products, duals, exact sequences, or the tensor unit. Such a realization does not by itself determine \(\operatorname{Aut}^{\otimes}(\omega)\). Tannakian symmetry begins only after the faithful exact tensor gate has been verified.
514 \end{boundary}
515
516 \begin{proposition}[Finite group model of the tensor boundary]
517 \label{prop:block12-finite-tensor-model}
518 Let \(G\) be finite and let \(\omega:\operatorname{Rep}_k(G)\to\operatorname{Vect}_k\) be the forgetful functor. The regular representation is faithful, the action on tensor products is diagonal, the unit is fixed, and the dual action is inverse transpose. These data distinguish every \(g\in G\) and realize the finite constant-group case of \cref{thm:block12-tannakian}.
519 \end{proposition}
520
521 \begin{proof}
522 Left translation on the regular representation is faithful. Functoriality of the tensor product, unit, and dual actions is immediate from the representation axioms. Since different group elements act differently on the regular basis, the induced tensor automorphisms are distinct.
523 \end{proof}
524
525 \begin{figure}[htbp]
526 \centering
527 \includegraphics[width=.97\textwidth]{galois_profinite_tannakian.pdf}
528 \caption{Finite quotient projectors converge strongly in profinite realizations. The lower row isolates the additional exact tensor structure required before a symmetry group scheme can be reconstructed.}
529 \label{fig:block12-profinite-tannakian}
530 \end{figure}
531
532 \subsection{The assembled finite Galois resolved atlas}
533 \label{subsec:block12-assembled}
534
535 The preceding results can be collected into one typed finite atlas. The reversible charts are:
536 \begin{enumerate}
537 \item subgroups \(H\leq G\), fixed fields \(L^H\), and faithful Reynolds projectors \(P_H\);
538 \item finite-dimensional \(K\)-vector spaces and semilinear \(L\)-spaces with effective descent data;
539 \item finite \(\acute{e}\)tale fibers and their continuous permutation actions;
540 \item neutral Tannakian categories equipped with a specified faithful exact tensor fiber functor.
541 \end{enumerate}
542 Directional or obstructed bridges include:
543 \begin{enumerate}
544 \item restriction and induction between subgroup charts, decomposed by Mackey packets;
545 \item passage from a nonnormal chart to its core or normal-closure envelope;
546 \item projectivization from linear to projective cocycles;
547 \item forgetting tensor structure from a Tannakian fiber functor to an additive or corner realization;
548 \item passage from finite quotients to a profinite inverse system.
549 \end{enumerate}
550
551 \begin{theorem}[Finite semisimple Galois and Tannakian assembly]
552 \label{thm:block12-galois-assembly}
553 Under the hypotheses printed in this section:
554 \begin{enumerate}
555 \item the subgroup, fixed-field, and faithful fixed-projector charts form an order-reversing normalized atlas;
556 \item normal charts admit quotient compression, while nonnormal charts form conjugate projector atlases with canonical core and normal-closure envelopes;
557 \item double cosets give exact transition packets for induction and restriction;
558 \item ramification filtrations give nested fixed projectors whose weighted unresolved ranks are the Artin and Swan conductors;
559 \item continuous finite-dimensional profinite representations are detected at a finite quotient level;
560 \item linear semilinear descent is effective on objects and morphisms;
561 \item projective descent is effective exactly when its relative Brauer class vanishes;
562 \item neutral Tannakian symmetry is reconstructed exactly when a faithful exact tensor fiber functor is supplied.
563 \end{enumerate}
564 These statements form a complete assembly theorem for the declared finite semisimple and neutral tensor strata. They do not imply effective descent for modular, wild, nonneutral, or arbitrary heterogeneous atlases.
565 \end{theorem}
566
567 \begin{proof}
568 Items 1--5 are \cref{thm:block12-normalized-galois-chart,thm:block12-normality,thm:block12-core-normal-closure,thm:block12-mackey,thm:block12-artin-conductor,cor:block12-finite-level}. Item 6 is \cref{thm:block12-vector-descent}. Item 7 is \cref{thm:block12-brauer}. Item 8 is \cref{thm:block12-tannakian}. The final boundary follows because the hypotheses used by Reynolds averaging, Hilbert 90, and neutral Tannakian reconstruction are not automatic in the excluded settings.
569 \end{proof}
570
571 A complete chart record on the finite subgroup locus is
572 \[
573 \mathcal J_{L/K}(H)
574 =
575 \left(
576 H,
577 L^H,
578 P_H,
579 \operatorname{Core}_G(H),
580 \operatorname{Ncl}_G(H),
581 \{K\backslash G/H\}_{K\leq G}
582 \right).
583 \]
584 The first three entries already classify the normalized object, while the remaining entries record its atlas position and transition structure. For a framed semilinear vector object, the descended \(K\)-space with its frame is complete. For a projective object, the Brauer class is an obstruction coordinate, not a complete invariant of all central simple algebras without the remaining algebra data.
585
586 \subsection{Exact finite verification}
587 \label{subsec:block12-verification}
588
589 The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, \(V_4\), \(S_3\), \(D_4\), and \(A_4\); every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commuting and noncommuting subgroup pairs; double-coset partitions and the Mackey dimension law; cyclic ramification-style filtrations; explicit Hilbert 90 coboundaries over \(\mathbb C/\mathbb R\); the split and quaternionic projective examples; and faithful regular tensor actions.
590
591 \input{generated/block12_validation_table.tex}
592
593 The verifier records \(11{,}195\) exact checks and no failures. Together with the preceding certified atlas program, the cumulative exact record is
594 \[
595 \boxed{2{,}038{,}860}
596 \]
597 checks with no recorded failures.
598
599 \begin{boundarybox}
600 \textbf{Exact boundary of Galois-Tannakian atlas.}
601 The finite subgroup/projector identities, semilinear vector descent, relative Brauer obstruction, profinite finite-level resolution, and neutral Tannakian tensor gate are established under their printed hypotheses. The block does not prove exactness of fixed points when the group order is not invertible, derived replacement of failed Reynolds projectors, integral wild descent, classification of all central simple algebras from the Brauer class alone, existence of a neutral fiber functor, or effective descent for the entire heterogeneous Komorebi atlas.
602 \end{boundarybox}
603
604 \begin{resultbox}
605 \textbf{Section result.}
606 Finite semisimple Galois theory now appears as a fully typed resolved stratum. Subgroups, fixed fields, and faithful fixed projectors are reversible charts. Nonnormality produces conjugate atlases and mixed Mackey transition packets. Ramification is a nested detail resolution. Hilbert 90 makes linear semilinear descent effective, while the relative Brauer class is the exact obstruction to projective vector descent. Profinite quotient charts reconstruct continuous finite-dimensional symmetry, and a faithful exact tensor fiber functor crosses the separate Tannakian boundary to reconstruct an affine group scheme. This closes the declared finite Galois and neutral tensor stratum, not modular, wild, nonneutral, or unrestricted global descent.
607 \end{resultbox}
Other PDFs for this work
Original file and source record
This copy matches the recorded source file. Its publication here does not change the work’s review status.
SHA-256: a198bc61ce484674e016e69b8bb8183d27cedbfc65d1ff9b57d3a14a3395fd70