Lemma 59.107.1. Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_ p$. Consider the sheaf $\mathcal{O}^{perf} = \mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}$ on $(\mathit{Sch}/S)_{fppf}$. Then $\mathcal{O}^{perf}$ is in the essential image of $R\epsilon _* : D((\mathit{Sch}/S)_ h) \to D((\mathit{Sch}/S)_{fppf})$.
59.107 Cohomology of the structure sheaf in the h topology
Let $p$ be a prime number. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site with $p\mathcal{O} = 0$. Then we set $\mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}$ equal to the colimit in the category of sheaves of rings of the system
\[ \mathcal{O} \xrightarrow {F} \mathcal{O} \xrightarrow {F} \mathcal{O} \xrightarrow {F} \ldots \]
where $F : \mathcal{O} \to \mathcal{O}$, $f \mapsto f^ p$ is the Frobenius endomorphism.
Proof. We prove this using the criterion of Lemma 59.106.3. Before check the conditions, we note that for a quasi-compact and quasi-separated object $X$ of $(\mathit{Sch}/S)_{fppf}$ we have
\[ H^ i_{fppf}(X, \mathcal{O}^{perf}) = \mathop{\mathrm{colim}}\nolimits _ F H^ i_{fppf}(X, \mathcal{O}) \]
See Cohomology on Sites, Lemma 21.16.1. We will also use that $H^ i_{fppf}(X, \mathcal{O}) = H^ i(X, \mathcal{O})$, see Descent, Proposition 35.9.3.
Let $A, f, J$ be as in More on Flatness, Example 38.37.10 and consider the associated almost blow up square. Since $X$, $X'$, $Z$, $E$ are affine, we have no higher cohomology of $\mathcal{O}$. Hence we only have to check that
\[ 0 \to \mathcal{O}^{perf}(X) \to \mathcal{O}^{perf}(X') \oplus \mathcal{O}^{perf}(Z) \to \mathcal{O}^{perf}(E) \to 0 \]
is a short exact sequence. This was shown in (the proof of) More on Flatness, Lemma 38.38.2.
Let $X, X', Z, E$ be as in More on Flatness, Example 38.37.11. Since $X$ and $Z$ are affine we have $H^ p(X, \mathcal{O}_ X) = H^ p(Z, \mathcal{O}_ X) = 0$ for $p > 0$. By More on Flatness, Lemma 38.38.1 we have $H^ p(X', \mathcal{O}_{X'}) = 0$ for $p > 0$. Since $E = \mathbf{P}^1_ Z$ and $Z$ is affine we also have $H^ p(E, \mathcal{O}_ E) = 0$ for $p > 0$. As in the previous paragraph we reduce to checking that
\[ 0 \to \mathcal{O}^{perf}(X) \to \mathcal{O}^{perf}(X') \oplus \mathcal{O}^{perf}(Z) \to \mathcal{O}^{perf}(E) \to 0 \]
is a short exact sequence. This was shown in (the proof of) More on Flatness, Lemma 38.38.2. $\square$
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 Here on the left hand side by $\mathcal{O}^ h$ we mean the h sheafification of the structure sheaf.
\[ H^ i((\mathit{Sch}/S)_ h, \mathcal{O}^ h) = \mathop{\mathrm{colim}}\nolimits _ F H^ i(S, \mathcal{O}) \]
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). $\square$
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