Proposition 59.107.2 (0EWU)—The Stacks project

Proposition 59.107.2. Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_ p$. Then

\[ H^ i((\mathit{Sch}/S)_ h, \mathcal{O}^ h) = \mathop{\mathrm{colim}}\nolimits _ F H^ i(S, \mathcal{O}) \]

Here on the left hand side by $\mathcal{O}^ h$ we mean the h sheafification of the structure sheaf.

Proof. This is just a reformulation of Lemma 59.107.1. Recall that $\mathcal{O}^ h = \mathcal{O}^{perf} = \mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}$, see More on Flatness, Lemma 38.38.7. By Lemma 59.107.1 we see that $\mathcal{O}^{perf}$ viewed as an object of $D((\mathit{Sch}/S)_{fppf})$ is of the form $R\epsilon _*K$ for some $K \in D((\mathit{Sch}/S)_ h)$. Then $K = \epsilon ^{-1}\mathcal{O}^{perf}$ which is actually equal to $\mathcal{O}^{perf}$ because $\mathcal{O}^{perf}$ is an h sheaf. See Cohomology on Sites, Section 21.27. Hence $R\epsilon _*\mathcal{O}^{perf} = \mathcal{O}^{perf}$ (with apologies for the confusing notation). Thus the lemma now follows from Leray

\[ R\Gamma _ h(S, \mathcal{O}^{perf}) = R\Gamma _{fppf}(S, R\epsilon _*\mathcal{O}^{perf}) = R\Gamma _{fppf}(S, \mathcal{O}^{perf}) \]

and the fact that

\[ H^ i_{fppf}(S, \mathcal{O}^{perf}) = H^ i_{fppf}(S, \mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}) = \mathop{\mathrm{colim}}\nolimits _ F H^ i_{fppf}(S, \mathcal{O}) \]

as $S$ is quasi-compact and quasi-separated (see proof of Lemma 59.107.1). $\square$

The Stacks project

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