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. $\square$

Comment #1473 by Xiaowen Hu on

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

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 $U_{i_0} \times_S \ldots \times_S U_{i_p}$ are affine since $U$ 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}.

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