The Stacks project

Lemma 59.100.5. Consider the comparison morphism $\epsilon : (\mathit{Sch}/S)_{fppf} \to (\mathit{Sch}/S)_{\acute{e}tale}$. Let $\mathcal{P}$ denote the class of finite morphisms of schemes. For $X$ in $(\mathit{Sch}/S)_{\acute{e}tale}$ denote $\mathcal{A}'_ X \subset \textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ the full subcategory consisting of sheaves of the form $\pi _ X^{-1}\mathcal{F}$ with $\mathcal{F}$ in $\textit{Ab}(X_{\acute{e}tale})$. Then Cohomology on Sites, Properties (1), (2), (3), (4), and (5) of Cohomology on Sites, Situation 21.30.1 hold.

Proof. We first show that $\mathcal{A}'_ X \subset \textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 12.10.3. Parts (1), (2), (3) are immediate as $\pi _ X^{-1}$ is exact and fully faithful for example by Lemma 59.99.4. If $0 \to \pi _ X^{-1}\mathcal{F} \to \mathcal{G} \to \pi _ X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ then $0 \to \mathcal{F} \to \pi _{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma 59.99.4. Hence $\mathcal{G} = \pi _ X^{-1}\pi _{X, *}\mathcal{G}$ is in $\mathcal{A}'_ X$ which checks the final condition.

Cohomology on Sites, Property (1) holds by the existence of fibre products of schemes and the fact that the base change of a finite morphism of schemes is a finite morphism of schemes, see Morphisms, Lemma 29.44.6.

Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 34.4.17.

Cohomology on Sites, Property (3) is Lemma 59.100.1.

Cohomology on Sites, Property (4) holds by Lemma 59.99.5 part (4).

Cohomology on Sites, Property (5) is implied by More on Morphisms, Lemma 37.48.1. $\square$

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