The Stacks project

68.7 Higher vanishing for quasi-coherent sheaves

In this section we show that given a quasi-compact and quasi-separated algebraic space $X$ there exists an integer $n = n(X)$ such that the cohomology of any quasi-coherent sheaf on $X$ vanishes beyond degree $n$.

Lemma 68.7.1. With $S$, $W$, $G$, $U$, $\chi $ as in Lemma 68.6.5. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, then so is $\mathcal{F} \otimes _{\mathbf{Z}} \underline{\mathbf{Z}}(\chi )$.

Proof. The $\mathcal{O}_ U$-module structure is clear. To check that $\mathcal{F} \otimes _{\mathbf{Z}} \underline{\mathbf{Z}}(\chi )$ is quasi-coherent it suffices to check étale locally. Hence the lemma follows as $\underline{\mathbf{Z}}(\chi )$ is finite locally free as a $\underline{\mathbf{Z}}$-module. $\square$

The following proposition is interesting even if $X$ is a scheme. It is the natural generalization of Cohomology of Schemes, Lemma 30.4.2. Before we state it, observe that given an étale morphism $f : U \to X$ from an affine scheme towards a quasi-separated algebraic space $X$ the fibres of $f$ are universally bounded, in particular there exists an integer $d$ such that the fibres of $|U| \to |X|$ all have size at most $d$; this is the implication $(\eta ) \Rightarrow (\delta )$ of Decent Spaces, Lemma 67.5.1.

Proposition 68.7.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and separated. Let $U$ be an affine scheme, and let $f : U \to X$ be a surjective étale morphism. Let $d$ be an upper bound for the size of the fibres of $|U| \to |X|$. Then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^ q(X, \mathcal{F}) = 0$ for $q \geq d$.

Proof. We will use the spectral sequence of Lemma 68.6.6. The lemma applies since $f$ is separated as $U$ is separated, see Morphisms of Spaces, Lemma 66.4.10. Since $X$ is separated the scheme $U \times _ X \ldots \times _ X U$ is a closed subscheme of $U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \ldots \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U$ hence is affine. Thus $W_ p$ is affine. Hence $U_ p = W_ p/S_{p + 1}$ is an affine scheme by Groupoids, Proposition 39.23.9. The discussion in Section 68.3 shows that cohomology of quasi-coherent sheaves on $W_ p$ (as an algebraic space) agrees with the cohomology of the corresponding quasi-coherent sheaf on the underlying affine scheme, hence vanishes in positive degrees by Cohomology of Schemes, Lemma 30.2.2. By Lemma 68.7.1 the sheaves $\mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)$ are quasi-coherent. Hence $H^ q(W_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero when $q > 0$. By our definition of the integer $d$ we see that $W_ p = \emptyset $ for $p \geq d$. Hence also $H^0(W_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero when $p \geq d$. This proves the proposition. $\square$

In the following lemma we establish that a quasi-compact and quasi-separated algebraic space has finite cohomological dimension for quasi-coherent modules. We are explicit about the bound only because we will use it later to prove a similar result for higher direct images.

Lemma 68.7.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Then we can choose

  1. an affine scheme $U$,

  2. a surjective étale morphism $f : U \to X$,

  3. an integer $d$ bounding the degrees of the fibres of $U \to X$,

  4. for every $p = 0, 1, \ldots , d$ a surjective étale morphism $V_ p \to U_ p$ from an affine scheme $V_ p$ where $U_ p$ is as in Lemma 68.6.6, and

  5. an integer $d_ p$ bounding the degree of the fibres of $V_ p \to U_ p$.

Moreover, whenever we have (1) – (5), then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^ q(X, \mathcal{F}) = 0$ for $q \geq \max (d_ p + p)$.

Proof. Since $X$ is quasi-compact we can find a surjective étale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma 65.6.3. By Decent Spaces, Lemma 67.5.1 the fibres of $f$ are universally bounded, hence we can find $d$. We have $U_ p = W_ p/S_{p + 1}$ and $W_ p \subset U \times _ X \ldots \times _ X U$ is open and closed. Since $X$ is quasi-separated the schemes $W_ p$ are quasi-compact, hence $U_ p$ is quasi-compact. Since $U$ is separated, the schemes $W_ p$ are separated, hence $U_ p$ is separated by (the absolute version of) Spaces, Lemma 64.14.5. By Properties of Spaces, Lemma 65.6.3 we can find the morphisms $V_ p \to W_ p$. By Decent Spaces, Lemma 67.5.1 we can find the integers $d_ p$.

At this point the proof uses the spectral sequence

\[ E_1^{p, q} = H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p)) \Rightarrow H^{p + q}(X, \mathcal{F}) \]

see Lemma 68.6.6. By definition of the integer $d$ we see that $U_ p = 0$ for $p \geq d$. By Proposition 68.7.2 and Lemma 68.7.1 we see that $H^ q(U_ p, \mathcal{F}|_{U_ p} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi _ p))$ is zero for $q \geq d_ p$ for $p = 0, \ldots , d$. Whence the lemma. $\square$


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