The Stacks project

Lemma 103.8.2. Let $\mathcal{X}$ be an algebraic stack.

  1. 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 in $\textit{LQCoh}^{fpc}(\mathcal{O}_{\mathcal{X}_ i})$, then $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

  2. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of flat and locally finitely presented 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 $\mathcal{X}_{fppf}$. If each $f_ j^{-1}\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_{\mathcal{X}_ i})$, then $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Proof. Part (1) follows from a combination of Lemmas 103.6.1 and 103.7.2. The proof of (2) is analogous to the proof of Lemma 103.6.3. Let $\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{fppf}$.

First, suppose there is a morphism $a : \mathcal{U} \to \mathcal{X}$ which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that $a^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Then there is an exact sequence

\[ 0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F} \]

where $b$ is the morphism $b : \mathcal{U} \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$, see Sheaves on Stacks, Proposition 96.19.7 and Lemma 96.19.10. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent and has the flat base change property, see Proposition 103.8.1. The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent and have the flat base change property by Proposition 103.8.1. We conclude that $\mathcal{F}$ is locally quasi-coherent and has the flat base change property by Proposition 103.8.1.

Choose a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By part (1) it suffices to show that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Again by part (1) it suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma 101.27.14 there exists an fppf covering $\{ a_ i : U_ i \to U\} $ such that each $x \circ a_ i$ factors through some $f_ j$. Hence the module $a_ i^*\mathcal{F}$ on $(\mathit{Sch}/U_ i)_{fppf}$ is locally quasi-coherent and has the flat base change property. After refining the covering we may assume $\{ U_ i \to U\} _{i = 1, \ldots , n}$ is a standard fppf covering. Then $x^*\mathcal{F}$ is an fppf module on $(\mathit{Sch}/U)_{fppf}$ whose pullback by the morphism $a : U_1 \amalg \ldots \amalg U_ n \to U$ is locally quasi-coherent and has the flat base change property. Hence by the previous paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property as desired. $\square$


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 07AQ. Beware of the difference between the letter 'O' and the digit '0'.