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Dimension bound for quasi-coherent cohomology on a Noetherian scheme

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a separated Noetherian scheme (Locally Noetherian and Noetherian schemes) whose underlying topological space is Noetherian of dimension $\dim X \leq d$ for an integer $d \geq 0$ (Noetherian topological spaces via ACC on opens or DCC on closed subsets, Chain dimension and the empty-space convention); this is the finite Krull dimension $d$ of the design's finite-dimensional setting, the chain dimension of the underlying space being the dimension used throughout this page. Then $H^{q} (X, F) = 0$ for every quasi-coherent $O_{X}$ -module $F$ (Quasi-coherent module on a scheme) and every integer $q > d$ , where $H^{q}$ is sheaf cohomology (Sheaf cohomology as right derived global sections) applied to the underlying sheaf of abelian groups of $F$ (Modules on a ringed space). The empty scheme has no positive cohomology. Separation is retained from the design although it is not used: the topological vanishing theorem quoted below needs only a Noetherian space of dimension at most $d$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and a separated Noetherian scheme $X$ whose underlying space is Noetherian with $\dim X \leq d$ , $d \geq 0$ .

[F1]

A scheme is Noetherian when it is locally Noetherian and quasi-compact, equivalently when it has a finite affine open cover by spectra of Noetherian rings; such a cover $X = U_{1} \cup \dots \cup U_{n}$ with $U_{i}$ affine and $Γ (U_{i}, O_{X})$ Noetherian may therefore be chosen. (Locally Noetherian and Noetherian schemes)

[F2]

Under AC, $Spec (R)$ is a Noetherian topological space for every Noetherian commutative ring $R$ . (The spectrum of a Noetherian ring is a Noetherian topological space)

[F3]

Under AC, every subspace of a Noetherian topological space is Noetherian; in particular the open subspaces $U_{i}$ occurring in a cover of $X$ are Noetherian as topological spaces. (Noetherian topological spaces via ACC on opens or DCC on closed subsets, Subspaces of a Noetherian space and its compact open subsets)

[F4]

For a Noetherian topological space $T$ the dimension $\dim T$ is the supremum of the lengths of strict chains of nonempty irreducible closed subsets, $\dim \emptyset = - \infty$ , and $\dim T$ may be infinite. (Chain dimension and the empty-space convention)

[F5]

Under AC, if $T$ is a Noetherian topological space with $\dim T \leq d$ for an integer $d \geq 0$ , then $H^{q} (T, G) = 0$ for every sheaf of abelian groups $G$ on $T$ and every integer $q > d$ . (Grothendieck vanishing on a Noetherian space)

[F6]

A quasi-coherent $O_{X}$ -module is in particular an $O_{X}$ -module, hence a sheaf of abelian groups on $X$ with $O_{X} (U)$ -module structures compatible with restriction; a morphism of $O_{X}$ -modules is in particular a morphism of abelian sheaves. (Quasi-coherent module on a scheme, Modules on a ringed space)

[F7]

Under AC the sheaf cohomology groups $H^{q} (X, G)$ of an abelian sheaf $G$ on a topological space $X$ are defined as the right derived objects of global sections relative to a fixed supplied injective resolution datum, and $H^{q} (X, G) = 0$ for $q < 0$ . (Sheaf cohomology as right derived global sections)

Proof

technique · direct: pass from the finite affine cover of the Noetherian scheme to Noetherianness of the underlying space, then apply the published topological vanishing theorem to the underlying abelian sheaf of the quasi-coherent module; the empty scheme is covered by the $\dim=-\infty$ convention

1.1F1

By [F1] there is a finite affine open cover $X = U_{1} \cup \dots \cup U_{n}$ with each $R_{i} : = Γ (U_{i}, O_{X})$ a Noetherian ring, where $n = 0$ occurs exactly when $X = \emptyset$ ; the underlying space of the affine scheme $U_{i}$ is $Spec (R_{i})$ with its Zariski topology.

1.2F2F5given

The Axiom of Choice is available as a hypothesis and is consumed exactly in the two quoted results that require it, the Noetherian-spectrum theorem [F2] and the topological vanishing theorem [F5].

2.1F2F3step 1.1

For each $i$ , $Spec (R_{i})$ is Noetherian by [F2], and $U_{i}$ is an open subspace of $X$ whose topology is that of $Spec (R_{i})$ , so each $U_{i}$ is a Noetherian topological space.

3.1F4step 1.1step 2.1construct

If $n \geq 1$ the space $X$ is Noetherian: for an ascending chain $V_{0} \subseteq V_{1} \subseteq \dots$ of open subsets of $X$ the restrictions $V_{j} \cap U_{i}$ form, for each fixed $i$ , an ascending chain of open subsets of the Noetherian space $U_{i}$ , hence stabilize for $j \geq j_{i}$ ; with $j_{0} : = \max_{i} j_{i}$ , which exists because the cover is finite, one has $V_{j} \cap U_{i} = V_{j_{0}} \cap U_{i}$ for all $j \geq j_{0}$ and all $i$ , and since the $U_{i}$ cover $X$ this gives $V_{j} = V_{j_{0}}$ for all $j \geq j_{0}$ . Thus every ascending chain of opens of $X$ stabilizes and the underlying space of $X$ is Noetherian; it is of dimension $\dim X \leq d$ by hypothesis.

4.1F4F5step 3.1given

If $n = 0$ then $X = \emptyset$ by step 1.1, and [F4] gives $\dim X = - \infty \leq d$ ; in either case $X$ is a Noetherian topological space of dimension at most $d$ , so [F5] applies and yields $H^{q} (X, G) = 0$ for every sheaf of abelian groups $G$ on $X$ and every integer $q > d$ .

5.1F6F7step 4.1∎

Let $F$ be a quasi-coherent $O_{X}$ -module. By [F6] its underlying sheaf of abelian groups is an abelian sheaf on $X$ , so step 4.1 applied to that sheaf gives $H^{q} (X, F) = 0$ for every $q > d$ , the symbol $H^{q} (X, F)$ denoting the group of [F7] for the underlying abelian sheaf. In particular every quasi-coherent $O_{X}$ -module has vanishing cohomology above degree $d$ , and for $X = \emptyset$ there is no positive-degree cohomology at all in the sense that every abelian sheaf on $X$ has zero groups in every positive degree, as follows from step 4.1 with any $d \geq 0$ .

Depends on

Used by

Serre duality for locally free sheaves on a smooth projective variety Theorem

Dependency tree · two levels

54 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, §§30.2–30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), §§19.1, 19.6, 19.9, 28.1–28.2 (standard reference, not scraped)