The Stacks project

Lemma 59.103.4. Consider the comparison morphism $\epsilon : (\mathit{Sch}/S)_ h \to (\mathit{Sch}/S)_{\acute{e}tale}$. Let $\mathcal{P}$ denote the class of proper morphisms. 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 21.30.1 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. 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, the fact that the base change of a proper morphism of schemes is a proper morphism of schemes, see Morphisms, Lemma 29.41.5, and the fact that the base change of a morphism of finite presentation is a morphism of finite presentation, see Morphisms, Lemma 29.21.4.

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

Cohomology on Sites, Property (3) is Lemma 59.103.1.

Cohomology on Sites, Property (4) holds by Lemma 59.99.5 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 59.78.2.

Cohomology on Sites, Property (5) is implied by More on Morphisms, Lemma 37.48.1 combined with the fact that a surjective finite locally free morphism is surjective, proper, and of finite presentation and hence defines a h-covering by More on Flatness, Lemma 38.34.7. $\square$

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