Lemma 103.8.2 (07AQ)—The Stacks project

Lemma 103.8.2. 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 in $\textit{LQCoh}^{fpc}(\mathcal{O}_{\mathcal{X}_ i})$, then $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.
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. $\square$

The Stacks project

Comments (0)

Post a comment