The Stacks project

Lemma 103.6.1. Let $\mathcal{X}$ be an algebraic stack. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_ j|(|\mathcal{X}_ j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{\acute{e}tale}$. If each $f_ j^{-1}\mathcal{F}$ is locally quasi-coherent, then so is $\mathcal{F}$.

Proof. We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 101.33.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.12.3. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective smooth morphism. Let $x$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x$, $x_ i$ live over schemes $V$, $V_ i$, that $\{ V_ i \to V\} $ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. The restriction $x_ i^*\mathcal{F}|_{V_{i, {\acute{e}tale}}}$ is equal to the restriction of $f^*\mathcal{F}$ to $V_{i, {\acute{e}tale}}$, see Sheaves on Stacks, Lemma 96.9.3. Hence $x^*\mathcal{F}|_{V_{\acute{e}tale}}$ is a sheaf on the small étale site of $V$ which is quasi-coherent when restricted to $V_{i, {\acute{e}tale}}$ for each $i$. This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma 66.29.6. $\square$


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Your email address will not be published.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 075Y. Beware of the difference between the letter 'O' and the digit '0'.