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The Stacks project

Lemma 103.6.1. 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 locally quasi-coherent, then so is \mathcal{F}.

Proof. We may replace each of the algebraic stacks \mathcal{X}_ j by a scheme U_ j (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 101.33.2). The pullback of \mathcal{F} to (\mathit{Sch}/U_ j)_{\acute{e}tale} is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.12.3. Then f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X} is a surjective smooth morphism. Let x be an object of \mathcal{X}. By Sheaves on Stacks, Lemma 96.19.10 there exists an étale covering \{ x_ i \to x\} _{i \in I} such that each x_ i lifts to an object u_ i of (\mathit{Sch}/U)_{\acute{e}tale}. This just means that x, x_ i live over schemes V, V_ i, that \{ V_ i \to V\} is an étale covering, and that x_ i comes from a morphism u_ i : V_ i \to U. The restriction x_ i^*\mathcal{F}|_{V_{i, {\acute{e}tale}}} is equal to the restriction of f^*\mathcal{F} to V_{i, {\acute{e}tale}}, see Sheaves on Stacks, Lemma 96.9.3. Hence x^*\mathcal{F}|_{V_{\acute{e}tale}} is a sheaf on the small étale site of V which is quasi-coherent when restricted to V_{i, {\acute{e}tale}} for each i. This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma 66.29.6. \square


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