Lemma 103.14.6. Let $\mathcal{X}$ be an algebraic stack.

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

$\mathcal{F}$ is parasitic, and

$g^*\mathcal{F} = 0$ 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 103.14.2.

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

$\mathcal{F}$ is parasitic, and

$g^*\mathcal{F} = 0$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is as in Lemma 103.14.2.

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