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Finite-dimensional coherent cohomology over a field

Statement

Assume the Axiom of Choice, inherited from the proper finiteness theorem and the basis-extraction corollary cited below (The Axiom of Choice). Let $k$ be a field (Field), let $X$ be a scheme proper over $k$ , that is, the structure morphism $X \to Spec k$ is proper (Proper morphisms), and let $F$ be a coherent $O_{X}$ -module (Coherent module sheaves). Then for every $q \geq 0$ the $k$ -vector space $H^{q} (X, F)$ of sheaf cohomology (Sheaf cohomology as right derived global sections) is finite-dimensional over $k$ (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis), and only finitely many of the groups $H^{q} (X, F)$ are nonzero. More precisely, choose a finite affine open cover of $X$ with no empty members when $X \neq \emptyset$ , and use the empty cover when $X = \emptyset$ . If $n$ is its number of members, then $H^{q} (X, F) = 0$ for every $q \geq n$ , so the vanishing bound is the length of the cover and does not depend on $F$ .

The empty source $X = \emptyset$ , the zero sheaf $F = 0$ , the degree $q = 0$ and every field $k$ are included.

Facts & Assumptions

Given: A field $k$ , a proper morphism $X \to Spec k$ and a coherent $O_{X}$ -module $F$ ; the Axiom of Choice is inherited from the cited suppliers.

[F1]

A field is a Noetherian ring: its only ideals are $(0)$ and the whole field, so every ideal is finitely generated. (A field has only the zero ideal and itself, hence is Noetherian)

[F2]

Finite generation and vanishing for proper coherent cohomology: for a Noetherian ring $A$ , a proper morphism $X \to Spec A$ and a coherent $F$ , each $H^{q} (X, F)$ is a finitely generated $A$ -module; choose a finite affine open cover with no empty members when $X$ is nonempty and use the empty cover when $X$ is empty, and let $n$ be its number of members. Then $H^{q} (X, F) = 0$ for every $q \geq n$ . The cited theorem assumes both AC and DC; AC in this corollary supplies DC by the choice-implication theorem. (Finite coherent cohomology for proper schemes, AC implies DC implies countable choice)

[F3]

A module over a field is a vector space: the axioms for a $k$ -module are exactly the vector-space axioms over $k$ , so every $H^{q} (X, F)$ is a $k$ -vector space. (Vector space over a field, Generated submodule, cyclic and finitely generated modules, module basis and free module)

[F4]

Finitely generated modules and finite spanning sets: a module $M$ over a ring is finitely generated when it is generated by finitely many of its elements, that is, when there are $m_{1}, \dots, m_{s} \in M$ with every element of $M$ a finite linear combination of the $m_{j}$ ; a set of vectors spans a vector space when its linear combinations fill it. Hence a finitely generated $k$ -module is a $k$ -vector space with a finite spanning set. (Generated submodule, cyclic and finitely generated modules, module basis and free module, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ )

[F5]

Basis extraction: assuming the Axiom of Choice, every spanning subset of a vector space contains a basis. (Every spanning subset of a vector space contains a basis)

[F6]

Finite-dimensionality: a vector space is finite-dimensional over its field when it has a finite basis, and the zero space is finite-dimensional of dimension $0$ via the empty basis. (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis)

[F7]

The Axiom of Choice is the choice principle named in the statement. (The Axiom of Choice)

Proof

technique · direct: specialise the proper finiteness theorem to the base field, where a finitely generated module is a vector space with a finite spanning set, extract a finite basis, and read off the finite vanishing range from the cover-length bound

1.1F1F2

The base is Noetherian. By [F1] the field $k$ is a Noetherian ring, and AC supplies the DC premise of [F2] by the choice-implication theorem cited there. If $X$ is empty, choose the empty cover; if $X$ is nonempty, choose a finite affine open cover and discard any empty members, leaving at least one member. Write $n$ for the number of members of this chosen cover. Thus [F2] applies to the proper morphism $X \to Spec k$ and the coherent module $F$ : each $H^{q} (X, F)$ is a finitely generated $k$ -module, and $H^{q} (X, F) = 0$ for every $q \geq n$ .

1.2F3F4F5F61.1

Finite-dimensionality in every degree. By [F3] each $H^{q} (X, F)$ is a $k$ -vector space, and by 1.1 it is finitely generated as a $k$ -module, hence has a finite spanning set by [F4]; [F5] produces a basis of $H^{q} (X, F)$ inside that finite spanning set, which is a finite basis, so $H^{q} (X, F)$ is finite-dimensional over $k$ by [F6].

1.3F21.1

Only finitely many nonzero groups. The cover chosen in 1.1 exists because $X$ is proper, hence quasi-compact, over the affine base $Spec k$ . By 1.1, $H^{q} (X, F) = 0$ for every $q \geq n$ , so the only degrees that can be nonzero are $q = 0, 1, \dots, n - 1$ , a finite set (empty when $X = \emptyset$ ).

2.1F2F4F5F6F71.11.2∎

Boundaries and choice accounting. If $X = \emptyset$ then $n = 0$ , all groups vanish by 1.1, and each is finite-dimensional of dimension $0$ by [F6]; if $F = 0$ all groups vanish in the same way. Degree $q = 0$ is covered by 1.2 together with all other degrees. The field $k$ is arbitrary, in particular $k = F_{2}$ and fields of every characteristic are allowed, and only the trivial ring $A = 0$ is excluded by the hypothesis that $k$ is a field. The Axiom of Choice [F7] supplies DC for the proper finiteness theorem through [F2] and is also used by the basis extraction of [F5], which is the only place where a basis is selected; the finite spanning sets of [F4] are supplied by 1.1, so no selection from an infinite family occurs.

Depends on

Used by

Dependency tree · two levels

83 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)