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Acyclicity on intersections of a standard affine cover

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative ring, let $X = Spec A$ (The underlying space of an affine spectrum), let $f_{1}, \dots, f_{r} \in A$ generate the unit ideal, so that the distinguished opens $D (f_{1}), \dots, D (f_{r})$ form a finite standard cover of $X$ , and let $F$ be a quasi-coherent $O_{X}$ -module (Quasi-coherent module on a scheme). For every nonempty subset $S \subseteq {1, \dots, r}$ put $g_{S} = \prod_{i \in S} f_{i}$ . Then $⋂_{i \in S} D (f_{i}) = D (g_{S}) ≅ Spec A_{g_{S}}$ as schemes, and $H^{q} (⋂_{i \in S} D (f_{i}), F ∣_{⋂_{i \in S} D (f_{i})}) = 0 for every q > 0,$ where $H^{q}$ is sheaf cohomology (Sheaf cohomology as right derived global sections).

The degenerate cases are included and are read through the same identities: if $g_{S}$ is nilpotent then $A_{g_{S}} = 0$ and the intersection is the empty scheme $Spec 0 = \emptyset$ , on which the restriction of $F$ is the zero sheaf; and the empty intersection, corresponding to $S = \emptyset$ , is $X = D (1) = Spec A_{1}$ itself, again an affine scheme with a quasi-coherent sheaf on it. The unit-ideal hypothesis is not used in the proof below; it is retained because the statement is consumed for standard covers.

Facts & Assumptions

Given: The Axiom of Choice, a commutative ring $A$ , elements $f_{1}, \dots, f_{r} \in A$ , and a quasi-coherent $O_{X}$ -module $F$ on $X = Spec A$ .

[F1]

For $h \in A$ the morphism induced by $A \to A_{h}$ identifies $Spec (A_{h})$ with the open locally ringed subspace $D (h)$ of $Spec A$ ; moreover $D (0) = \emptyset$ , $D (1) = Spec A$ and $D (h k) = D (h) \cap D (k)$ for all $h, k \in A$ . (A principal localization identifies its spectrum with a distinguished open, Distinguished-subset identities)

[F2]

If $F$ is quasi-coherent on a scheme $X$ and $V \subseteq X$ is open, then the restriction $F ∣_{V}$ is a quasi-coherent $O_{V}$ -module. (Quasi-coherent module on a scheme)

[F3]

If $X = Spec B$ is an affine scheme, including the empty affine scheme and the zero ring, and $G$ is a quasi-coherent $O_{X}$ -module, then $H^{q} (X, G) = 0$ for every $q > 0$ . (Affine acyclicity of quasi-coherent sheaves)

[F4]

Sheaf cohomology is the right derived functor of global sections: an isomorphism of ringed spaces $φ : Y \to X$ induces an equivalence of abelian-sheaf categories carrying $O_{Y}$ -modules to $O_{X}$ -modules and identifies the global-sections functors, so that $H^{q} (Y, φ^{*} G) ≅ H^{q} (X, G)$ for every $O_{X}$ -module $G$ ; also $H^{q} = 0$ for $q < 0$ by convention. (Sheaf cohomology as right derived global sections)

Proof

technique · direct: identify each finite intersection of distinguished opens with a principal distinguished open, transport to the affine scheme $\operatorname{Spec}A_{g_S}$, and apply affine acyclicity

1.1F1

Let $S = {i_{1}, \dots, i_{k}} \subseteq {1, \dots, r}$ be nonempty. Iterating the identity $D (h) \cap D (k) = D (h k)$ of [F1] gives $⋂_{i \in S} D (f_{i}) = D (g_{S})$ with $g_{S} = \prod_{i \in S} f_{i}$ , and [F1] applied to $h = g_{S}$ identifies this open locally ringed subspace with $Spec A_{g_{S}}$ .

1.2F1

Degenerate cases. If $S = \emptyset$ then the empty intersection is $X = D (1) = Spec A_{1}$ by [F1]. If $g_{S}$ is nilpotent for some nonempty $S$ , then $A_{g_{S}} = 0$ and $D (g_{S}) = \emptyset = Spec 0$ by [F1], and the restriction of $F$ to the empty scheme is the zero sheaf.

2.1F2F3F4step 1.1

Let $S$ be nonempty with intersection $Y = D (g_{S}) \neq \emptyset$ . Then $Y$ is an open subscheme of $X$ , so $F ∣_{Y}$ is a quasi-coherent $O_{Y}$ -module by [F2]; under the isomorphism $Y ≅ Spec A_{g_{S}}$ of step 1.1 it corresponds to a quasi-coherent module $G$ on the affine scheme $Spec A_{g_{S}}$ , and [F4] identifies $H^{q} (Y, F ∣_{Y}) ≅ H^{q} (Spec A_{g_{S}}, G)$ . By [F3] the right-hand group vanishes for every $q > 0$ , hence so does the left-hand group.

2.2F3F4step 1.1step 1.2

Let $S$ be nonempty with $Y = D (g_{S}) = \emptyset$ . Then $A_{g_{S}} = 0$ by step 1.2, the restriction of $F$ to $Y$ is the zero sheaf on the empty ringed space, and $H^{q} (Y, F ∣_{Y}) = 0$ for every $q > 0$ : this is the empty affine scheme case of [F3] transported along the isomorphism $Y ≅ Spec 0$ of step 1.1, or directly the case of the zero sheaf.

2.3F3step 1.2

For $S = \emptyset$ the empty intersection is $X = Spec A$ by step 1.2, an affine scheme with the quasi-coherent module $F$ , so $H^{q} (X, F) = 0$ for every $q > 0$ directly by [F3].

3.1F2F3F4step 1.1step 1.2step 2.1step 2.2step 2.3∎

Combining the cases, for every nonempty $S \subseteq {1, \dots, r}$ the intersection $⋂_{i \in S} D (f_{i}) = D (g_{S})$ is isomorphic to $Spec A_{g_{S}}$ and the restriction of $F$ to it has vanishing cohomology in every positive degree. The argument used only that each finite intersection of distinguished opens is a principal distinguished open, hence an affine open subscheme possibly empty; the unit-ideal hypothesis on $f_{1}, \dots, f_{r}$ was not needed, and the quasi-coherence of the restrictions was supplied by [F2]. The Axiom of Choice is inherited from [F3] and [F4] and no additional selection is made.

Depends on

Used by

Higher direct images localize over an affine base Lemma

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2 (standard reference, not scraped)