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Restricting an associated sheaf to a localization

Example

Assume the Axiom of Choice, inherited from the associated-sheaf construction. Let $A$ be a commutative ring with $1$ , let $f \in A$ , let $M$ be an $A$ -module, and let $σ : Spec A_{f} ⟶ D (f) \subseteq Spec A$ be the isomorphism of locally ringed spaces induced by the localisation $A \to A_{f}$ (A principal localization identifies its spectrum with a distinguished open). Write $\tilde{M}$ for the associated sheaf of $M$ on $Spec A$ and $\tilde{M_{f}}$ for the associated sheaf of the $A_{f}$ -module $M_{f}$ on $Spec A_{f}$ (The associated module sheaf exists).

Then $σ$ identifies $\tilde{M_{f}}$ with the restriction of $\tilde{M}$ to $D (f)$ : there is a canonical isomorphism of $O_{Spec A_{f}}$ -modules $\tilde{M_{f}} ≅ σ^{*} (\tilde{M} ∣_{D (f)}),$ natural in $M$ and in $f$ . On corresponding basic opens the identification is the canonical one: for $a \in A$ the open $D_{A_{f}} (a / 1) \subseteq Spec A_{f}$ satisfies $σ (D_{A_{f}} (a / 1)) = D (f a) \subseteq D (f)$ , and the sections are $M_{f a}$ on both sides. In particular the example includes the degenerate case where $f$ is nilpotent, when both sides are the zero sheaf on the empty scheme.

Facts & Assumptions

Given: The Axiom of Choice; a commutative ring $A$ ; an element $f \in A$ ; an $A$ -module $M$ ; the isomorphism $σ : Spec A_{f} \to D (f)$ .

[F1]

The localisation $A \to A_{f}$ induces an isomorphism of locally ringed spaces $Spec A_{f} \to D (f)$ onto the open subscheme $D (f)$ , whose underlying map sends a prime of $A_{f}$ to its contraction in $A$ (A principal localization identifies its spectrum with a distinguished open).

[F2]

For an affine scheme $Spec B$ with associated sheaf $\tilde{N}$ , one has $Γ (D (b), \tilde{N}) = N_{b}$ for every $b \in B$ , with restriction the canonical localisation, naturally in $N$ and $b$ (Sections of the associated sheaf on basic opens, The associated module sheaf exists).

[F3]

For an affine open $W = Spec C \subseteq Spec A$ with inclusion $σ$ and corresponding ring map $φ : A \to C$ , and any $A$ -module $M$ , there is a canonical isomorphism $(\tilde{M}) ∣_{W} ≅ \tilde{(C \otimes_{A} M)}$ of $O_{W}$ -modules, natural in $M$ (An associated sheaf restricts to an associated sheaf on an affine open).

[F4]

For the localisation $A \to A_{f}$ and an $A$ -module $M$ one has $A_{f} \otimes_{A} M ≅ M_{f}$ canonically, and further localisation $(M_{f})_{a / 1} ≅ M_{f a}$ for $a \in A$ (Localisation of a module at a multiplicative subset).

[F5]

The refuted situation the example corrects: the restriction of $\tilde{M}$ to $D (f)$ is the associated sheaf of the localised module $M_{f}$ , not merely a sheaf with isomorphic stalks.

Proof technique: direct; identify the open immersion, apply the affine-open restriction theorem, and check the identification on basic opens.

Proof

1.1F1

By [F1] the map $σ$ is an isomorphism of locally ringed spaces onto $D (f)$ and a prime $q \subseteq A_{f}$ corresponds to $q \cap A$ ; hence for $a \in A$ the basic open $D_{A_{f}} (a / 1)$ corresponds to $D_{A} (a) \cap D_{A} (f) = D_{A} (f a)$ , and the sections of the structure sheaf on these corresponding opens are the same ring $A_{f a}$ under $σ$ .

1.2F3F4

Applying [F3] with $W = D (f) = Spec A_{f}$ , $C = A_{f}$ and the ring map $A \to A_{f}$ gives a canonical isomorphism $(\tilde{M}) ∣_{D (f)} ≅ \tilde{(A_{f} \otimes_{A} M)}$ of $O_{D (f)}$ -modules, natural in $M$ ; by [F4] $A_{f} \otimes_{A} M ≅ M_{f}$ , so transporting along $σ$ gives the canonical isomorphism $\tilde{M_{f}} ≅ σ^{*} (\tilde{M} ∣_{D (f)})$ of the Example.

2.1F2F4step 1.1step 1.2

The identification is the canonical one on basic opens: for $a \in A$ , [F2] applied to $Spec A$ gives $Γ (D (f a), \tilde{M}) = M_{f a}$ , while [F2] applied to $Spec A_{f}$ with the element $a / 1$ gives $Γ (D_{A_{f}} (a / 1), \tilde{M_{f}}) = (M_{f})_{a / 1}$ , and [F4] identifies $(M_{f})_{a / 1} ≅ M_{f a}$ with the localisation of the identity on $M$ ; for $b \in A$ with $D (f b) \subseteq D (f a)$ the restriction maps are the canonical localisations $M_{f a} \to M_{f b}$ and $(M_{f})_{a / 1} \to (M_{f})_{b / 1}$ , which [F4] identifies, so these identifications realise the isomorphism of step 1.2 on a basis of $D (f)$ and are natural in $M$ and $f$ .

3.1F2F3step 1.2step 2.1∎

If $f$ is nilpotent then $D (f) = \emptyset$ and $A_{f} = 0$ , so $M_{f} = 0$ and both sides of the isomorphism are the zero sheaf on the empty scheme, in agreement with [F2]; otherwise the isomorphism of step 1.2 is the restriction identification of the Example, and the Axiom of Choice is inherited from [F2] and [F3], no new choice being made.

Depends on

Used by

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Sources

The Stacks Project, Schemes, §§26.5, 26.7, 26.24 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17 (standard reference, not scraped)