The Stacks project

Lemma 82.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. Then $\pi _ X^{-1}\mathcal{F}$ is given by the rule

\[ (\pi _ X^{-1}\mathcal{F})(Y) = \Gamma (Y_{\acute{e}tale}, f_{small}^{-1}\mathcal{F}) \]

for $f : Y \to X$ in $(\textit{Spaces}/X)_{\acute{e}tale}$. Moreover, $\pi _ Y^{-1}\mathcal{F}$ satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.

Proof. Since pullback is transitive and $f_{small} = \pi _ X \circ i_ f$ (see above) we see that $i_ f^{-1} \pi _ X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$. This shows that $\pi _ X^{-1}$ has the description given in the lemma.

To prove that $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma 71.8.7 to show that for a surjective proper morphism $V \to U$ of algebraic spaces over $X$ we have $(\pi _ X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps $(\pi _ X^{-1}\mathcal{F})(V) \to (\pi _ X^{-1}\mathcal{F})(V \times _ U V)$. This we have seen in Lemma 82.4.1.

The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma 71.8.2.

Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be an fpqc covering of algebraic spaces over $X$. Let $s_ i \in (\pi _ X^{-1}\mathcal{F})(U_ i)$ be sections which agree over $U_ i \times _ U U_ j$. We have to prove there exists a unique $s \in (\pi _ X^{-1}\mathcal{F})(U)$ restricting to $s_ i$ over $U_ i$. Case I: $U$ and $U_ i$ are schemes. This case follows from Étale Cohomology, Lemma 58.39.2. Case II: $U$ is a scheme. Here we choose surjective étale morphisms $T_ i \to U_ i$ where $T_ i$ is a scheme. Then $\mathcal{T} = \{ T_ i \to U\} $ is an fpqc covering by schemes and by case I the result holds for $\mathcal{T}$. We omit the verification that this implies the result for $\mathcal{U}$. Case III: general case. Let $W \to U$ be a surjective étale morphism, where $W$ is a scheme. Then $\mathcal{W} = \{ U_ i \times _ U W \to W\} $ is an fpqc covering (by algebraic spaces) of the scheme $W$. By case II the result hold for $\mathcal{W}$. We omit the verification that this implies the result for $\mathcal{U}$. $\square$


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Your email address will not be published.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 0DG7. Beware of the difference between the letter 'O' and the digit '0'.