Lemma 96.23.1. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal {X}$-modules. Assume
$\mathcal{F}$ is locally quasi-coherent, and
for any morphism $\varphi : x \to y$ of $\mathcal{X}$ which lies over a morphism of schemes $f : U \to V$ which is flat and locally of finite presentation the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_{\acute{e}tale}} \to \mathcal{F}|_{U_{\acute{e}tale}}$ of (96.9.4.1) is an isomorphism.
Then $\mathcal{F}$ is a sheaf for the fppf topology.
Proof.
Let $\{ x_ i \to x\} $ be an fppf covering of $\mathcal{X}$ lying over the fppf covering $\{ f_ i : U_ i \to U\} $ of schemes over $S$. By assumption the restriction $\mathcal{G} = \mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent and the comparison maps $f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, {\acute{e}tale}}}$ are isomorphisms. Hence the sheaf condition for $\mathcal{F}$ and the covering $\{ x_ i \to x\} $ is equivalent to the sheaf condition for $\mathcal{G}^ a$ on $(\mathit{Sch}/U)_{fppf}$ and the covering $\{ U_ i \to U\} $ which holds by Descent, Lemma 35.8.1.
$\square$
Comments (0)