The Stacks project

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. $\square$


Comments (2)

Comment #1473 by Xiaowen Hu on

I think in the following part of the proof, all capital should be .

Next, assume that is separated. Choose an affine open covering of . The family is then an fpqc covering, and all the intersections are affine since is separated. So all rows of the spectral sequence of Theorem \ref{theorem-cech-ss} are zero, except the zeroth row. Therefore where the last equality results from standard scheme theory, see Cohomology of Schemes, Lemma \ref{coherent-lemma-cech-cohomology-quasi-coherent}.

There are also:

  • 2 comment(s) on Section 56.22: Cohomology of quasi-coherent sheaves

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03P2. Beware of the difference between the letter 'O' and the digit '0'.