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Projective n-space has quasi-coherent cohomological dimension at most n

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative ring and let $n \geq 0$ be an integer, and let $P_{A}^{n}$ be the projective space over $A$ with its standard charts (Relative projective space from standard charts). Then every quasi-coherent $O_{P_{A}^{n}}$ -module $F$ (Quasi-coherent module on a scheme) satisfies $H^{q} (P_{A}^{n}, F) = 0$ for every integer $q > n$ , 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 case $n = 0$ is included; the zero ring $A = 0$ is allowed, and then $P_{A}^{n} = \emptyset$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a commutative ring $A$ , an integer $n \geq 0$ , and a quasi-coherent $O_{P_{A}^{n}}$ -module $F$ .

[F1]

Chart model of projective space: $P_{A}^{n} = Spec A \times_{Spec Z} P_{Z}^{n}$ (Relative projective space from standard charts, Existence of all scheme fibre products). Its standard charts are $U_{i} = Spec B_{i}$ with $B_{i} = A [x_{ℓ}^{(i)} : ℓ \neq i]$ , they form an open cover of $P_{A}^{n}$ , each $U_{i}$ is affine, and for $i \neq j$ the transition isomorphism of the gluing identifies $U_{i} \cap U_{j}$ with the distinguished open subset $D (x_{j}^{(i)}) \subseteq U_{i}$ with ring $(B_{i})_{x_{j}^{(i)}}$ , the two transition formulas being reciprocal; on $U_{i} \cap U_{j}$ the inverse identification $B_{j} \to (B_{i})_{x_{j}^{(i)}}$ is given by $x_{m}^{(j)} \mapsto x_{m}^{(i)} / x_{j}^{(i)}$ for $m \neq i, j$ and $x_{i}^{(j)} \mapsto 1 / x_{j}^{(i)}$ . The charts and their overlaps commute with base change, and when $A = 0$ all $B_{i}$ are the zero ring, so all charts are empty.

[F2]

Distinguished opens: for $f \in R$ the set $D (f) = {p \in Spec R : f \notin p}$ (Principal distinguished subsets of the prime spectrum) satisfies $D (f) \cap D (g) = D (f g)$ , the morphism induced by $R \to R_{f}$ identifies $Spec (R_{f})$ with the open subscheme $D (f)$ of $Spec R$ (A principal localization identifies its spectrum with a distinguished open, The underlying space of an affine spectrum), and $Γ (D (f), O) = R_{f}$ with restrictions the canonical localisation maps (Sections and restrictions on distinguished opens of an affine scheme).

[F3]

Quasi-compactness: the structure morphism $π : P_{A}^{n} \to Spec A$ is of finite type (Projective space is of finite type over its base), and a finite-type morphism is by definition quasi-compact (Locally finite type and finite type morphisms). For a quasi-compact morphism the inverse image of every affine open of the target is quasi-compact (Quasi-compactness is local on the target and survives base change), and every affine scheme is quasi-compact (Every affine scheme is quasi-compact, Quasi-compact and quasi-separated schemes).

[F4]

Separatedness gluing criterion: for a morphism $f : X \to S$ , an affine open cover $S = ⋃_{i} W_{i} = ⋃_{i} Spec A_{i}$ and affine open covers $f^{- 1} (W_{i}) = ⋃_{j} U_{i j} = ⋃_{j} Spec B_{i j}$ , the morphism $f$ is separated if and only if for all $i, j, k$ the intersection $U_{i j} \cap U_{i k}$ is affine and the natural ring map $B_{i j} \otimes_{A_{i}} B_{i k} \to Γ (U_{i j} \cap U_{i k}, O_{X})$ is surjective; empty intersections use the zero ring. Separatedness of $X \to Spec Z$ means that the scheme $X$ is separated (Affine-overlap criterion for separatedness, Separated morphism of schemes, Schemes and morphisms over a base).

[F5]

Čech comparison for separated schemes: under the Axiom of Choice, for a quasi-compact separated scheme $X$ , a finite affine open cover $U_{0}, \dots, U_{r}$ of $X$ and a quasi-coherent $O_{X}$ -module $F$ , every intersection of one or more members of the cover is affine and the canonical comparison map ${\overset{ˇ}{H}}^{q} (U, F) \to H^{q} (X, F)$ is an isomorphism for every $q \geq 0$ (Cech cohomology computes quasi-coherent cohomology on a separated scheme).

[F6]

Ordered Čech complex of a finite cover: for a cover $U = (U_{i})_{i \in I}$ indexed by a linearly ordered set, $C^{q} (U, F) = \prod_{i_{0} < \dots < i_{q}} F (U_{i_{0}} \cap \dots \cap U_{i_{q}})$ , and when $I$ has no increasing $(q + 1)$ -tuple this product is empty and $C^{q} (U, F) = 0$ ; the ordered Čech cohomology ${\overset{ˇ}{H}}^{q} (U, F)$ is the cohomology of this complex (Ordered Čech cochain complex of a cover, Fixed-cover Čech cohomology).

Proof

technique · direct: the $n+1$ standard affine charts of $\mathbb P^n_A$ cover the scheme, all their finite intersections are distinguished opens in a chart, the gluing criterion proves separatedness over $\operatorname{Spec}\mathbb Z$, and the comparison theorem identifies sheaf cohomology with the Čech cohomology of that finite cover, whose complex is concentrated in degrees $0,\dots,n$

1.1F1

Set $X = P_{A}^{n}$ and let $U = (U_{0}, \dots, U_{n})$ be the ordered family of standard charts, with $U_{i} = Spec B_{i}$ and $B_{i} = A [x_{ℓ}^{(i)} : ℓ \neq i]$ . By [F1] each $U_{i}$ is an open affine subscheme and the $U_{i}$ cover $X$ ; in particular $X$ is a scheme over $Spec Z$ .

1.2F1F2

Every finite intersection of distinct members $U_{i_{0}}, \dots, U_{i_{m}}$ of $U$ is affine: inside the affine chart $U_{i_{0}}$ it is $⋂_{ℓ = 1}^{m} D (x_{i_{ℓ}}^{(i_{0})}) = D (x_{i_{1}}^{(i_{0})} \dots x_{i_{m}}^{(i_{0})})$ by [F1] and [F2], and $D (f)$ is affine with ring $(B_{i_{0}})_{f}$ by [F2]. For $m = 0$ this is the affine chart $U_{i_{0}}$ itself.

1.3F3

$X$ is quasi-compact: $π : X \to Spec A$ is of finite type, hence quasi-compact, so $X = π^{- 1} (Spec A)$ is quasi-compact by [F3].

1.4F1F2F4

$X$ is separated as a scheme. Apply the criterion [F4] to the structure morphism $X \to Spec Z$ , the single affine open $W = Spec Z$ of the base and the affine cover ${U_{i}}$ of $X$ : for $j = k$ the ring $U_{j}$ is affine and $B_{j} \otimes_{Z} B_{j} \to Γ (U_{j}, O_{X}) = B_{j}$ is the multiplication map, which is surjective; for $j \neq k$ the intersection $U_{j} \cap U_{k}$ is affine by 1.2, $Γ (U_{j} \cap U_{k}, O_{X}) = (B_{j})_{x_{k}^{(j)}}$ by [F1] and [F2], and the natural map $B_{j} \otimes_{Z} B_{k} \to (B_{j})_{x_{k}^{(j)}}$ sends $b \otimes 1 \mapsto b$ and, by the reciprocal transition formula of [F1], $1 \otimes x_{m}^{(k)} \mapsto x_{m}^{(j)} / x_{k}^{(j)}$ for $m \neq j, k$ and $1 \otimes x_{j}^{(k)} \mapsto 1 / x_{k}^{(j)}$ ; its image therefore contains $B_{j}$ and the inverse $1 / x_{k}^{(j)}$ , hence is the whole localisation $(B_{j})_{x_{k}^{(j)}} = B_{j} [1 / x_{k}^{(j)}]$ . The criterion gives that $X \to Spec Z$ is separated.

1.5F51.21.31.4

By [F5], applied to the quasi-compact separated scheme $X$ , the finite affine open cover $U_{0}, \dots, U_{n}$ and the quasi-coherent module $F$ , the canonical comparison map ${\overset{ˇ}{H}}^{q} (U, F) \to H^{q} (X, F)$ is an isomorphism for every $q \geq 0$ .

1.6F61.5

For $q > n$ the index set ${0, \dots, n}$ of the ordered cover $U$ has no increasing $(q + 1)$ -tuple, so $C^{q} (U, F) = 0$ by [F6] and hence ${\overset{ˇ}{H}}^{q} (U, F) = 0$ . By the isomorphism of 1.5, $H^{q} (X, F) = 0$ for every $q > n$ .

2.1F1F5F61.31.6∎

Boundary and choice accounting. For $n = 0$ the cover is the single chart $U_{0} = Spec A$ , only $C^{0}$ is nonzero, and the conclusion $H^{q} = 0$ for $q > 0$ follows from 1.6 (it also follows from the affinity of $X = Spec A$ ). For $A = 0$ the ring $B_{i}$ is the zero ring, $U_{i} = Spec 0 = \emptyset$ , so $X = \emptyset$ by [F1] and ${U_{0}, \dots, U_{n}}$ is still an affine open cover of $X$ ; the same argument applies verbatim, and $C^{q} = 0$ for all $q$ by [F6] since $F$ vanishes on the empty scheme. The case $F = 0$ is trivially included. Nothing is asserted for $q \leq n$ . The Axiom of Choice is a hypothesis, consumed exactly through the Čech comparison theorem [F5] (and the sheaf-cohomology framework it refers to); the remaining steps 1.1-1.4 and 1.6 make no choice, the indexes $0, \dots, n$ being finite and the gluing data being fixed by the definition.

Depends on

Used by

Dependency tree · two levels

65 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)