The Stacks project

Lemma 35.12.4. Let $S$ be a scheme. Let $\tau \in \{ Zar, {\acute{e}tale}\} $. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_{fppf}$ such that

  1. $\mathcal{G}|_{S_\tau }$ is quasi-coherent, and

  2. for every flat, locally finitely presented morphism $g : U \to S$ the canonical map $g_{\tau , small}^*(\mathcal{G}|_{S_\tau }) \to \mathcal{G}|_{U_\tau }$ is an isomorphism.

Then $H^ p(U, \mathcal{G}) = H^ p(U, \mathcal{G}|_{U_\tau })$ for every $U$ flat and locally of finite presentation over $S$.

Proof. Let $\mathcal{F}$ be the pullback of $\mathcal{G}|_{S_\tau }$ to the big fppf site $(\mathit{Sch}/S)_{fppf}$. Note that $\mathcal{F}$ is quasi-coherent. There is a canonical comparison map $\varphi : \mathcal{F} \to \mathcal{G}$ which by assumptions (1) and (2) induces an isomorphism $\mathcal{F}|_{U_\tau } \to \mathcal{G}|_{U_\tau }$ for all $g : U \to S$ flat and locally of finite presentation. Hence in the short exact sequences

\[ 0 \to \mathop{\mathrm{Ker}}(\varphi ) \to \mathcal{F} \to \mathop{\mathrm{Im}}(\varphi ) \to 0 \]


\[ 0 \to \mathop{\mathrm{Im}}(\varphi ) \to \mathcal{G} \to \mathop{\mathrm{Coker}}(\varphi ) \to 0 \]

the sheaves $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are parasitic for the fppf topology. By Lemma 35.12.2 we conclude that $H^ p(U, \mathcal{F}) \to H^ p(U, \mathcal{G})$ is an isomorphism for $g : U \to S$ flat and locally of finite presentation. Since the result holds for $\mathcal{F}$ by Proposition 35.9.3 we win. $\square$

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