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Projective coherent finiteness and large twist vanishing

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a Noetherian commutative ring, let $n \geq 0$ , let $X = P_{A}^{n}$ with twisting sheaves $O_{X} (m)$ (Twisting sheaf on Proj, Relative projective space from standard charts) and let $G$ be a coherent $O_{X}$ -module (Coherent module sheaves). Then:

$H^{q} (X, G)$ is a finite $A$ -module for every $q \geq 0$ ;
for every $q > 0$ there is an integer $m_{0} = m_{0} (G, q)$ such that $H^{q} (X, G (m)) = 0$ for all $m \geq m_{0}$ , where $G (m) = G \otimes_{O_{X}} O_{X} (m)$ .

The same two conclusions hold for a closed subscheme $i : X ↪ P_{A}^{n}$ and a coherent $O_{X}$ -module $G$ , with $O_{X} (1) = i^{*} O_{P_{A}^{n}} (1)$ and $G (m) = G \otimes_{O_{X}} O_{X} (m)$ . The zero module, the zero ring $A = 0$ and the case $n = 0$ are included.

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian commutative ring $A$ , a coherent sheaf $G$ on $P_{A}^{n}$ (and, for the last clause, a closed subscheme $i : X ↪ P_{A}^{n}$ with a coherent $G$ on $X$ ).

[F1]

For Noetherian $A$ the scheme $P_{A}^{n}$ is locally Noetherian: the standard affine charts are spectra of polynomial rings in finitely many variables over $A$ , which are Noetherian rings, and $P_{A}^{n}$ is quasi-compact as it is covered by the $n + 1$ standard charts. On a locally Noetherian scheme a quasi-coherent module is coherent exactly when it is of finite type, and kernels, cokernels, images and extensions of coherent modules are coherent. (Projective space is Proj of a polynomial ring, Relative projective space from standard charts, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves, Finite type and finitely presented module sheaves)

[F2]

The support ${x : H_{x} \neq 0}$ of a quasi-coherent module of finite type is closed; in particular a finite-type quasi-coherent module with zero stalk at a point vanishes on an open neighbourhood of that point. (Support of a finite-type quasi-coherent sheaf is closed)

[F3]

The identity closed immersion $P_{A}^{n} \to P_{A}^{n}$ exhibits $O (1)$ as H-very ample relative to the affine base; the projection is quasi-compact by its finite standard affine cover, so $O (1)$ is ample by Relative very ampleness implies relative ampleness and Relative very ampleness in the finite projective-space convention. Global generation and eventual generation: $H$ is globally generated when its evaluation map $Γ (X, H) \otimes_{Z} O_{X} \to H$ is surjective, and for a coherent $G$ on $P_{A}^{n}$ projective over the Noetherian ring $A$ the twist $G (m)$ is globally generated for all $m \geq m_{1} (G)$ . (Global generation by the evaluation map, Eventual generation of coherent projective twists)

[F4]

Twists: $O_{X} (m)$ is invertible, with $O_{X} (m) \otimes O_{X} (- m) ≅ O_{X}$ ; tensoring by an invertible module is exact and preserves coherence, because on an open cover the twisting module is trivial and exactness and coherence are local. The cohomology of twists on projective space is known: for every $d$ , $H^{q} (P_{A}^{n}, O (d)) = 0$ unless $q = 0$ or $q = n$ , with $H^{0} (P_{A}^{n}, O (d))$ free of finite rank for every $d$ , and it is zero for $d < 0$ when $n \geq 1$ ; for $n = 0$ it is $A$ for every $d$ . The top group $H^{n} (P_{A}^{n}, O (d))$ is free of finite rank (possibly zero) for every $d$ ; moreover $H^{q} (P_{A}^{n}, F) = 0$ for every $q > n$ and every quasi-coherent $F$ . (Invertible sheaves, Dual of a line bundle is its tensor inverse, Cohomology of O(d) on projective space, Projective n-space has quasi-coherent cohomological dimension at most n, Exact sequences of sheaves, The direct sum of an indexed family of modules)

[F5]

The long exact sequence: a short exact sequence of sheaves of modules $0 \to A \to B \to C \to 0$ on a scheme yields an exact sequence of abelian groups $\dots \to H^{q} (X, A) \to H^{q} (X, B) \to H^{q} (X, C) \to H^{q + 1} (X, A) \to \dots$ . (Long exact sequence of sheaf cohomology)

[F6]

Over a Noetherian ring, submodules and quotient modules of finite modules are finite. (Noetherian commutative rings and modules)

[F7]

Closed immersions preserve cohomology and coherence of pushforwards: for a closed immersion $i : X \to Y$ and a quasi-coherent $G$ on $X$ one has $H^{q} (X, G) ≅ H^{q} (Y, i_{*} G)$ for all $q \geq 0$ , and if $Y$ is locally Noetherian and $G$ is coherent then $i_{*} G$ is coherent. (Closed immersion preserves cohomology and coherent pushforward)

[F8]

A map $O_{X}^{\oplus N} \to H$ is surjective if and only if its cokernel is zero, and the cokernel of a map of quasi-coherent finite-type modules is quasi-coherent of finite type; its support is closed by [F2]. (Kernels and cokernels of quasi-coherent modules)

[F9]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct: use global generation of a high twist to present a coherent sheaf as a quotient of a finite sum of line bundles, then run two descending inductions on the cohomological degree, one for vanishing of high twists and one for finiteness; transport the result along a closed immersion for the subscheme clause

1.1F1F4

Setup. Since $A$ is Noetherian, $X = P_{A}^{n}$ is locally Noetherian and quasi-compact by [F1]. Fix a coherent $G$ ; by [F1] it is quasi-coherent of finite type, and so are all its twists by [F4].

1.2F4

Base of both inductions. By [F4] one has $H^{q} (X, F) = 0$ for every $q > n$ and every quasi-coherent $F$ , in particular for coherent ones; this is the base case $q = n + 1$ of both descending inductions below.

1.3F1F2F3F4F8

A finite presentation by line bundles. By [F3] there is $m_{1}$ such that $G (m_{1})$ is globally generated, so the evaluation map $Γ (X, G (m_{1})) \otimes_{Z} O_{X} \to G (m_{1})$ is surjective. For each point $x$ choose finitely many global sections generating the finitely generated stalk $G (m_{1})_{x}$ ; the cokernel of the induced map $O_{X}^{\oplus N_{x}} \to G (m_{1})$ is quasi-coherent of finite type with zero stalk at $x$ , hence vanishes on an open neighbourhood of $x$ by [F2] and [F8], and finitely many such neighbourhoods cover the quasi-compact space $X$ by [F1]. The union of the corresponding finite sets of sections gives $N < \infty$ and a surjection $p : O_{X}^{\oplus N} \to G (m_{1})$ ; twisting by the invertible module $O_{X} (- m_{1})$ is exact and preserves coherence by [F4], so it yields an exact sequence $0 ⟶ K ⟶ E \overset{q}{\to} G ⟶ 0, E = O_{X} (- m_{1})^{\oplus N},$ with $K$ coherent by [F1] and [F4].

2.1F4F5step 1.2step 1.3

Vanishing of high twists in positive degrees. We prove by descending induction on $q$ the statement $V (q)$ : for every coherent $F$ there is $m_{0} (F, q)$ with $H^{q} (X, F (m)) = 0$ for all $m \geq m_{0} (F, q)$ . For $q > n$ this holds with $m_{0} = 0$ for every $F$ by 1.2. Assume $q \geq 1$ and $V (q + 1)$ . Apply 1.3 to $F$ : there are $a \geq 0$ and a coherent $K$ with an exact sequence $0 \to K \to O_{X} (- a)^{\oplus N} \to F \to 0$ ; twisting by $O_{X} (m)$ and applying [F5] gives the exact segment $H^{q} (X, O_{X} (m - a))^{\oplus N} ⟶ H^{q} (X, F (m)) ⟶ H^{q + 1} (X, K (m)) ⟶ H^{q + 1} (X, O_{X} (m - a))^{\oplus N} .$ By [F4], $H^{q} (X, O_{X} (m - a)) = 0$ for every $m$ when $1 \leq q < n$ , and for $q = n$ when $m - a \geq - n$ ; for $q > n$ it vanishes by 1.2. Likewise $H^{q + 1} (X, O_{X} (m - a)) = 0$ for $m$ large: $q + 1 \geq 2$ , and if $q + 1 < n$ the group vanishes for all $m$ , if $q + 1 = n$ it vanishes for $m - a \geq - n$ , and if $q + 1 > n$ it vanishes by 1.2. Hence for $m$ large the outer groups vanish and exactness gives $H^{q} (X, F (m)) ≅ H^{q + 1} (X, K (m))$ , which is zero for $m \geq m_{0} (K, q + 1)$ by $V (q + 1)$ . Thus $V (q)$ holds, and with 1.2 the induction proves (2) for every $q > 0$ .

2.2F4F5F6step 1.2step 1.3

Finiteness. We prove by descending induction on $q$ the statement $F (q)$ : $H^{q} (X, F)$ is a finite $A$ -module for every coherent $F$ . For $q > n$ this is 1.2. Assume $q \leq n$ and $F (q + 1)$ , and apply 1.3 to $F$ to get $0 \to K \to O_{X} (- a)^{\oplus N} \to F \to 0$ . By [F4] each $H^{q} (X, O_{X} (- a))$ and each $H^{q + 1} (X, O_{X} (- a))$ is a finite free $A$ -module (possibly zero), so $H^{q}$ and $H^{q + 1}$ of $E = O_{X} (- a)^{\oplus N}$ are finite $A$ -modules, being finite direct sums; and $H^{q + 1} (X, K)$ is finite by $F (q + 1)$ . The exact segment of [F5], $H^{q} (X, E) \to H^{q} (X, F) \to H^{q + 1} (X, K) \to H^{q + 1} (X, E),$ gives a short exact sequence from the image of $H^{q} (X, E)$ in $H^{q} (X, F)$ to $H^{q} (X, F)$ and then to the kernel of $H^{q + 1} (X, K) \to H^{q + 1} (X, E)$ . The first term is a quotient of the finite module $H^{q} (X, E)$ , and the last is a submodule of the finite module $H^{q + 1} (X, K)$ , hence both are finite by [F6]. Lifting finite generators of the last term and adjoining finite generators of the first proves that $H^{q} (X, F)$ is finite. The induction gives (1) for every $q \geq 0$ .

3.1F7step 2.1step 2.2

The closed subscheme clause. Let $i : X ↪ P_{A}^{n}$ be a closed immersion and $G$ coherent on $X$ . By [F7] the pushforward $i_{*} G$ is coherent on $P_{A}^{n}$ and $H^{q} (X, G) ≅ H^{q} (P_{A}^{n}, i_{*} G)$ for all $q$ , so (1) for $G$ follows from 2.2 applied to $i_{*} G$ . For the twists, fix $m$ ; the identity $i_{*} (G (m)) ≅ (i_{*} G) (m)$ holds, because on an affine open $U = Spec R$ of $P_{A}^{n}$ contained in a standard chart the twisting module $O (1)$ is free of rank one, so $i^{- 1} (U)$ also lies where $i^{*} O (1)$ is free; writing $i^{- 1} (U) = Spec (R / I)$ one has $Γ (i^{- 1} U, G (m)) = Γ (i^{- 1} U, G)$ and $(i_{*} G) (m) (U) = Γ (i^{- 1} U, G) \otimes_{R} R$ , which agree, and on principal opens $D (f) \subseteq U$ both sides are the corresponding localisation with the same restriction maps; since the opens $U$ contained in standard charts form a basis, the two sheaves are equal. Hence $H^{q} (X, G (m)) ≅ H^{q} (P_{A}^{n}, (i_{*} G) (m))$ , which vanishes for $q > 0$ and $m$ large by 2.1 applied to the coherent sheaf $i_{*} G$ .

4.1F1F4F6F9step 1.3step 2.1∎

Boundaries and choice accounting. If $G = 0$ then all groups vanish and $m_{0} = 0$ works. If $A = 0$ then $P_{A}^{n} = \emptyset$ and $X = \emptyset$ for the closed subscheme clause, all groups are zero and finite over the zero ring, and the statements hold vacuously. If $n = 0$ then $X = Spec A$ is affine and $H^{q} = 0$ for $q > 0$ by [F4]; degree zero is $Γ (X, G)$ , a finite $A$ -module for coherent $G$ by [F1] and [F6] (for a closed subscheme of $Spec A$ the pushforward is a finite module over the Noetherian ring). For $q = n \geq 1$ the threshold in 2.1 includes the constraint $m \geq a - n$ from $H^{n} (O_{X} (m - a))$ , so finitely many small twists may be nonzero and the threshold is not effective. The Axiom of Choice [F9] is consumed through the affine charts, the finite generating selections in 1.3 and the long exact sequence; the inductions use no further choice of ideals or resolutions.

Depends on

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136 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, Sections 30.15-30.16 (Tags 0B5S, 01XS) (standard reference, not scraped)
Robin Hartshorne, Algebraic Geometry, Theorem III.5.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Section 28.2 (standard reference, not scraped)