The Stacks project

Lemma 102.14.6. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module with the flat base change property on $\mathcal{X}_{\acute{e}tale}$. The following are equivalent

    1. $\mathcal{F}$ is parasitic, and

    2. $g^*\mathcal{F} = 0$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ is as in Lemma 102.14.2.

  2. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module on $\mathcal{X}_{fppf}$. The following are equivalent

    1. $\mathcal{F}$ is parasitic, and

    2. $g^*\mathcal{F} = 0$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is as in Lemma 102.14.2.

Proof. Part (2) is immediate from the definitions (this is one of the advantages of the flat-fppf site over the lisse-étale site). The implication (1)(a) $\Rightarrow $ (1)(b) is immediate as well. To see (1)(b) $\Rightarrow $ (1)(a) let $U$ be a scheme and let $x : U \to \mathcal{X}$ be a surjective smooth morphism. Then $x$ is an object of the lisse-étale site of $\mathcal{X}$. Hence we see that (1)(b) implies that $\mathcal{F}|_{U_{\acute{e}tale}} = 0$. Let $V \to \mathcal{X}$ be an flat morphism where $V$ is a scheme. Set $W = U \times _\mathcal {X} V$ and consider the diagram

\[ \xymatrix{ W \ar[d]_ p \ar[r]_ q & V \ar[d] \\ U \ar[r] & \mathcal{X} } \]

Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_{\acute{e}tale}} \cong q_{small}^*\mathcal{F}|_{V_{\acute{e}tale}}$. Thus if $\mathcal{F}$ is in the kernel of $g^*$, then $\mathcal{F}|_{V_{\acute{e}tale}} = 0$ as desired. $\square$


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