Paper XII: Transport through an isomorphism R equals T

7 Transport through an isomorphism \(R\cong{\mathbb T}\)

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An isomorphism between a deformation ring and a Hecke algebra is an exact coordinate change. It transports support and finite presentation data, but it does not create a supported module.

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Theorem 7.1: Support and Fitting transport

Let \(\varphi:R\to{\mathbb T}\) be an isomorphism of Noetherian rings. Let \(M\) be a finite \({\mathbb T}\) module, viewed as an \(R\) module through \(\varphi\). Then the homeomorphism

\[ \varphi^*:{\operatorname{Spec}}{\mathbb T}\longrightarrow{\operatorname{Spec}} R \]

satisfies

\[ \boxed{ \varphi^*\bigl({\operatorname{Supp}}_{{\mathbb T}}(M)\bigr) = {\operatorname{Supp}}_R(M). } \]

For every \(j\geq0\),

\[ \boxed{ \varphi \left( {\operatorname{Fitt}}_j^R(M) \right) = {\operatorname{Fitt}}_j^{{\mathbb T}}(M). } \]

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Proof of Theorem 7.1

Localization commutes with transport along an isomorphism, so a localization of \(M\) vanishes on one side exactly when the corresponding localization vanishes on the other. This proves the support identity. A finite presentation matrix over \(R\) is carried entry by entry to a presentation matrix over \({\mathbb T}\). The minors generating each Fitting ideal are therefore carried to the corresponding minors.

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Corollary 7.2: Full support is invariant under \(R\cong{\mathbb T}\)

In the setting of Theorem 7.1, \(M\) has full \(R\) support if and only if it has full \({\mathbb T}\) support.

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Proof of Corollary 7.2

Apply the support homeomorphism in Theorem 7.1.

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Proposition 7.3: A quotient can erase a component

Let

\[ R=k[[x,y]]/(xy) \]

and let

\[ {\mathbb T}=R/(x)\cong k[[y]]. \]

Viewed as an \(R\) module, \({\mathbb T}\) has support \(V(x)\), which misses the generic point \((y)\) of the other irreducible component. Thus a surjection \(R\twoheadrightarrow{\mathbb T}\) does not by itself imply full component occupancy.

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Proof of Proposition 7.3

The annihilator of the \(R\) module \({\mathbb T}\) is \((x)\). Hence its support is \(V(x)\) by Theorem 6.2. The prime \((y)\) does not contain \(x\), so it is not in this support.

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Theorem 7.4: Field fiber criterion

Let \(R\) be a commutative ring, let \(M\) be a finite \(R\) module, and let \(x:R\to E'\) be a homomorphism to a field. Put \({\mathfrak p}=\ker x\). Then

\[ \boxed{ M\otimes_{R,x}E'\neq0 \quad\Longleftrightarrow\quad {\mathfrak p}\in{\operatorname{Supp}}_R(M). } \]

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Proof of Theorem 7.4

The map \(x\) factors through the local ring \(R_{{\mathfrak p}}\) and then through an embedding of the residue field \(\kappa({\mathfrak p})\) into \(E'\). Since \(M\) is finite, Nakayama's lemma gives

\[ M_{{\mathfrak p}}\neq0 \quad\Longleftrightarrow\quad M\otimes_R\kappa({\mathfrak p})\neq0. \]

Extension of scalars from \(\kappa({\mathfrak p})\) to \(E'\) preserves nonzero vector spaces.

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Corollary 7.5: Supported deformation points give nonzero Hecke fibers

Let \(R\cong{\mathbb T}\), and let \(M\) be a finite Hecke module. If a point \(x:R\to E'\) lies in \({\operatorname{Supp}}_R(M)\), then the corresponding Hecke fiber

\[ M\otimes_{{\mathbb T},x}E' \]

is nonzero.

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Proof of Corollary 7.5

Transport the support through Theorem 7.1, then apply Theorem 7.4.

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An isomorphism transports occupied support exactly. A quotient can remove a component.

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