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Lemma 58.96.4. Consider the comparison morphism $\epsilon : (\mathit{Sch}/S)_{ph} \to (\mathit{Sch}/S)_{\acute{e}tale}$. Let $\mathcal{P}$ denote the class of proper 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 $\pi _ X^{-1}\mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$ Then Cohomology on Sites, Properties (1), (2), (3), (4), and (5) of Cohomology on Sites, Situation 21.29.1 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 58.93.4. 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 58.93.4. In particular we see that $\pi _{X, *}\mathcal{G}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$. 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 proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma 29.41.5.

Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 34.4.16.

Cohomology on Sites, Property (3) is Lemma 58.96.1.

Cohomology on Sites, Property (4) holds by Lemma 58.93.5 part (2) and the fact that $R^ if_{small}\mathcal{F}$ is torsion if $\mathcal{F}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$, see Lemma 58.76.2.

Cohomology on Sites, Property (5) follows from More on Morphisms, Lemma 37.44.1 combined with the fact that a finite morphism is proper and a surjective proper morphism defines a ph covering, see Topologies, Lemma 34.8.6. $\square$


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