The Stacks project

Lemma 56.94.5. 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 21.29.1 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 56.93.4. 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 56.93.4. 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 28.42.6.

Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 33.4.16.

Cohomology on Sites, Property (3) is Lemma 56.94.1.

Cohomology on Sites, Property (4) holds by Lemma 56.93.5 part (4).

Cohomology on Sites, Property (5) is implied by More on Morphisms, Lemma 36.43.1. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F0H. Beware of the difference between the letter 'O' and the digit '0'.