The Stacks project

56.22 Cohomology of quasi-coherent sheaves

We start with a simple lemma (which holds in greater generality than stated). It says that the Čech complex of a standard covering is equal to the Čech complex of an fpqc covering of the form $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $ with $A \to B$ faithfully flat.

Lemma 56.22.1. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $(\mathit{Sch}/S)_\tau $, or on $S_\tau $ in case $\tau = {\acute{e}tale}$, and let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a standard $\tau $-covering of this site. Let $V = \coprod _{i \in I} U_ i$. Then

  1. $V$ is an affine scheme,

  2. $\mathcal{V} = \{ V \to U\} $ is a $\tau $-covering and an fpqc covering,

  3. the Čech complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ and $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ agree.

Proof. As the covering is a standard $\tau $-covering each of the schemes $U_ i$ is affine and $I$ is a finite set. Hence $V$ is an affine scheme. It is clear that $V \to U$ is flat and surjective, hence $\mathcal{V}$ is an fpqc covering, see Example 56.15.3. Note that $\mathcal{U}$ is a refinement of $\mathcal{V}$ and hence there is a map of Čech complexes $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$, see Cohomology on Sites, Equation (21.8.2.1). Next, we observe that if $T = \coprod _{j \in J} T_ j$ is a disjoint union of schemes in the site on which $\mathcal{F}$ is defined then the family of morphisms with fixed target $\{ T_ j \to T\} _{j \in J}$ is a Zariski covering, and so

56.22.1.1
\begin{equation} \label{etale-cohomology-equation-sheaf-coprod} \mathcal{F}(T) = \mathcal{F}(\coprod \nolimits _{j \in J} T_ j) = \prod \nolimits _{j \in J} \mathcal{F}(T_ j) \end{equation}

by the sheaf condition of $\mathcal{F}$. This implies the map of Čech complexes above is an isomorphism in each degree because

\[ V \times _ U \ldots \times _ U V = \coprod \nolimits _{i_0, \ldots i_ p} U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \]

as schemes. $\square$

Note that Equality (56.22.1.1) is false for a general presheaf. Even for sheaves it does not hold on any site, since coproducts may not lead to coverings, and may not be disjoint. But it does for all the usual ones (at least all the ones we will study).

Remark 56.22.2. In the statement of Lemma 56.22.1 the covering $\mathcal{U}$ is a refinement of $\mathcal{V}$ but not the other way around. Coverings of the form $\{ V \to U\} $ do not form an initial subcategory of the category of all coverings of $U$. Yet it is still true that we can compute Čech cohomology $\check H^ n(U, \mathcal{F})$ (which is defined as the colimit over the opposite of the category of coverings $\mathcal{U}$ of $U$ of the Čech cohomology groups of $\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings $\{ V \to U\} $. We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it.

Lemma 56.22.3 (Locality of cohomology). Let $\mathcal{C}$ be a site, $\mathcal{F}$ an abelian sheaf on $\mathcal{C}$, $U$ an object of $\mathcal{C}$, $p > 0$ an integer and $\xi \in H^ p(U, \mathcal{F})$. Then there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of $U$ in $\mathcal{C}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$.

Proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then $\xi $ is represented by a cocycle $\tilde{\xi } \in \mathcal{I}^ p(U)$ with $d^ p(\tilde{\xi }) = 0$. By assumption, the sequence $\mathcal{I}^{p - 1} \to \mathcal{I}^ p \to \mathcal{I}^{p + 1}$ in exact in $\textit{Ab}(\mathcal{C})$, which means that there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ such that $\tilde{\xi }|_{U_ i} = d^{p - 1}(\xi _ i)$ for some $\xi _ i \in \mathcal{I}^{p-1}(U_ i)$. Since the cohomology class $\xi |_{U_ i}$ is represented by the cocycle $\tilde{\xi }|_{U_ i}$ which is a coboundary, it vanishes. For more details see Cohomology on Sites, Lemma 21.7.3. $\square$

Theorem 56.22.4. Let $S$ be a scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ S$-module. Let $\mathcal{C}$ be either $(\mathit{Sch}/S)_\tau $ for $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $ or $S_{\acute{e}tale}$. Then

\[ H^ p(S, \mathcal{F}) = H^ p_\tau (S, \mathcal{F}^ a) \]

for all $p \geq 0$ where

  1. the left hand side indicates the usual cohomology of the sheaf $\mathcal{F}$ on the underlying topological space of the scheme $S$, and

  2. the right hand side indicates cohomology of the abelian sheaf $\mathcal{F}^ a$ (see Proposition 56.17.1) on the site $\mathcal{C}$.

Proof. We are going to show that $H^ p(U, f^*\mathcal{F}) = H^ p_\tau (U, \mathcal{F}^ a)$ for any object $f : U \to S$ of the site $\mathcal{C}$. The result is true for $p = 0$ by the sheaf property.

Assume that $U$ is affine. Then we want to prove that $H^ p_\tau (U, \mathcal{F}^ a) = 0$ for all $p > 0$. We use induction on $p$.

  1. Pick $\xi \in H^1_\tau (U, \mathcal{F}^ a)$. By Lemma 56.22.3, there exists an fpqc covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$. Up to refining $\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard $\tau $-covering. Applying the spectral sequence of Theorem 56.19.2, we see that $\xi $ comes from a cohomology class $\check\xi \in \check H^1(\mathcal{U}, \mathcal{F}^ a)$. Consider the covering $\mathcal{V} = \{ \coprod _{i\in I} U_ i \to U\} $. By Lemma 56.22.1, $\check H^\bullet (\mathcal{U}, \mathcal{F}^ a) = \check H^\bullet (\mathcal{V}, \mathcal{F}^ a)$. On the other hand, since $\mathcal{V}$ is a covering of the form $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $ and $f^*\mathcal{F} = \widetilde{M}$ for some $A$-module $M$, we see the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ is none other than the complex $(B/A)_\bullet \otimes _ A M$. Now by Lemma 56.16.4, $H^ p((B/A)_\bullet \otimes _ A M) = 0$ for $p > 0$, hence $\check\xi = 0$ and so $\xi = 0$.

  2. Pick $\xi \in H^ p_\tau (U, \mathcal{F}^ a)$. By Lemma 56.22.3, there exists an fpqc covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$. Up to refining $\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard $\tau $-covering. We apply the spectral sequence of Theorem 56.19.2. Observe that the intersections $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ are affine, so that by induction hypothesis the cohomology groups

    \[ E_2^{p, q} = \check H^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}^ a)) \]

    vanish for all $0 < q < p$. We see that $\xi $ must come from a $\check\xi \in \check H^ p(\mathcal{U}, \mathcal{F}^ a)$. Replacing $\mathcal{U}$ with the covering $\mathcal{V}$ containing only one morphism and using Lemma 56.16.4 again, we see that the Čech cohomology class $\check\xi $ must be zero, hence $\xi = 0$.

Next, assume that $U$ is separated. Choose an affine open covering $U = \bigcup _{i \in I} U_ i$ of $U$. The family $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ is then an fpqc covering, and all the intersections $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ are affine since $U$ is separated. So all rows of the spectral sequence of Theorem 56.19.2 are zero, except the zeroth row. Therefore

\[ H^ p_\tau (U, \mathcal{F}^ a) = \check H^ p(\mathcal{U}, \mathcal{F}^ a) = \check H^ p(\mathcal{U}, \mathcal{F}) = H^ p(U, \mathcal{F}) \]

where the last equality results from standard scheme theory, see Cohomology of Schemes, Lemma 29.2.6.

The general case is technical and (to extend the proof as given here) requires a discussion about maps of spectral sequences, so we won't treat it. It follows from Descent, Proposition 34.8.10 (whose proof takes a slightly different approach) combined with Cohomology on Sites, Lemma 21.7.1. $\square$

Remark 56.22.5. Comment on Theorem 56.22.4. Since $S$ is a final object in the category $\mathcal{C}$, the cohomology groups on the right-hand side are merely the right derived functors of the global sections functor. In fact the proof shows that $H^ p(U, f^*\mathcal{F}) = H^ p_\tau (U, \mathcal{F}^ a)$ for any object $f : U \to S$ of the site $\mathcal{C}$.


Comments (2)

Comment #4587 by Shiro on

In lemma 56.22.1, for the Zariski covering case, the covering may not be an open immersion. I wonder what's the precise meaning of covering in a big Zariski topos?

Comment #4589 by on

Dear Shiro, you are very right and thanks very much for pointing this out. The statement of the lemma has to be modified for the case of the Zariski topology because isn't a Zariski covering in that case (and only in that case -- the other cases are fine). All the definitions and conventions regarding Zariski coverings are given in Section 33.3. I will fix this the next time I go through all the comments.


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