The Stacks project

Lemma 84.8.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. For $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$.

  2. For $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ torsion we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

Proof. We have $a_ X^{-1}\mathcal{F} = \epsilon _ X^{-1} \pi _ X^{-1}\mathcal{F}$. By Lemma 84.5.1 the étale sheaf $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the ph topology and therefore is equal to $a_ X^{-1}\mathcal{F}$ (as pulling back by $\epsilon _ X$ is given by ph sheafification). Recall moreover that $\epsilon _{X, *}$ is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of $\pi _ X^{-1}$ in Lemma 84.5.1.

We will prove part (2) by reducing it to the case of schemes – see part (1) of Étale Cohomology, Lemma 59.102.5. This will “clearly work” as every algebraic space is étale locally a scheme. The details are given below but we urge the reader to skip the proof.

For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$ the higher direct image $R^ p\epsilon _{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^ p_{ph}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. Since every object of $(\textit{Spaces}/X)_{\acute{e}tale}$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^ p_{ph}(U, a_ X^{-1}\mathcal{F})$ we can find an étale covering $\{ U_ i \to U\} $ such that $\xi $ restricts to zero on $U_ i$. We have

\begin{align*} H^ p_{ph}(U, a_ X^{-1}\mathcal{F}) & = H^ p((\textit{Spaces}/U)_{ph}, (a_ X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\ & = H^ p((\mathit{Sch}/U)_{ph}, (a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U}) \end{align*}

where the second identification is Lemma 84.3.1 and the first is a general fact about restriction (Cohomology on Sites, Lemma 21.7.1). Looking at the first paragraph and the corresponding result in the case of schemes (Étale Cohomology, Lemma 59.102.1) we conclude that the sheaf $(a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U}$ matches the pullback by the “schemes version of $a_ U$”. Therefore we can find an étale covering $\{ U_ i \to U\} $ such that our class dies in $H^ p((\mathit{Sch}/U_ i)_{ph}, (a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U_ i})$ for each $i$, see Étale Cohomology, Lemma 59.102.5 (the precise statement one should use here is that $V_ n$ holds for all $n$ which is the statement of part (2) for the case of schemes). Transporting back (using the same formulas as above but now for $U_ i$) we conclude $\xi $ restricts to zero over $U_ i$ as desired. $\square$

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