The Stacks project

Lemma 46.5.8. Let $S$ be a scheme. Let $\mathcal{F}$ be an adequate $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $.

  1. The restriction $\mathcal{F}|_{S_{Zar}}$ is a quasi-coherent $\mathcal{O}_ S$-module on the scheme $S$.

  2. The restriction $\mathcal{F}|_{S_{\acute{e}tale}}$ is the quasi-coherent module associated to $\mathcal{F}|_{S_{Zar}}$.

  3. For any affine scheme $U$ over $S$ we have $H^ q(U, \mathcal{F}) = 0$ for all $q > 0$.

  4. There is a canonical isomorphism

    \[ H^ q(S, \mathcal{F}|_{S_{Zar}}) = H^ q((\mathit{Sch}/S)_\tau , \mathcal{F}). \]

Proof. By Lemma 46.3.5 and Lemma 46.5.2 we see that for any flat morphism of affines $U \to V$ over $S$ we have $\mathcal{F}(U) = \mathcal{F}(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U)$. This works in particular if $U \subset V \subset S$ are affine opens of $S$, hence $\mathcal{F}|_{S_{Zar}}$ is quasi-coherent. Thus (1) holds.

Let $S' \to S$ be an ├ętale morphism of schemes. Then for $U \subset S'$ affine open mapping into an affine open $V \subset S$ we see that $\mathcal{F}(U) = \mathcal{F}(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U)$ because $U \to V$ is ├ętale, hence flat. Therefore $\mathcal{F}|_{S'_{Zar}}$ is the pullback of $\mathcal{F}|_{S_{Zar}}$. This proves (2).

We are going to apply Cohomology on Sites, Lemma 21.10.9 to the site $(\mathit{Sch}/S)_\tau $ with $\mathcal{B}$ the set of affine schemes over $S$ and $\text{Cov}$ the set of standard affine $\tau $-coverings. Assumption (3) of the lemma is satisfied by Descent, Lemma 35.9.1 and Lemma 46.5.6 for the case of a covering by a single affine. Hence we conclude that $H^ p(U, \mathcal{F}) = 0$ for every affine scheme $U$ over $S$. This proves (3). In exactly the same way as in the proof of Descent, Proposition 35.9.3 this implies the equality of cohomologies (4). $\square$

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