The Stacks project

Lemma 102.4.2. Let $\mathcal{X}$ be an algebraic stack. Let $I$ be a set and for $i \in I$ let $x_ i : U_ i \to \mathcal{X}$ be an object of $\mathcal{X}$. Assume that $x_ i$ is flat and $\coprod x_ i : \coprod U_ i \to \mathcal{X}$ is surjective. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an arrow of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Denote $\varphi _ i$ the restriction of $\varphi $ to $(U_ i)_{\acute{e}tale}$. Then $\varphi $ is injective, resp. surjective, resp. an isomorphism if and only if each $\varphi _ i$ is so.

Proof. Choose a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. We may and do think of $x$ as an object of $\mathcal{X}$. This produces a presentation $\mathcal{X} = [U/R]$ for some groupoid in spaces $(U, R, s, t, c)$ and correspondingly an equivalence

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \mathit{QCoh}(U, R, s, t, c) \]

See discussion in Sheaves on Stacks, Section 95.15. The structure of abelian category on the right hand is such that $\varphi $ is injective, resp. surjective, resp. an isomorphism if and only if the restriction $\varphi |_{U_{\acute{e}tale}}$ is so, see Groupoids in Spaces, Lemma 77.12.6.

For each $i$ we choose an ├ętale covering $\{ W_{i, j} \to V \times _\mathcal {X} U_ i\} _{j \in J_ i}$ by schemes. Denote $g_{i, j} : W_{i, j} \to V$ and $h_{i, j} : W_{i, j} \to U_ i$ the obvious arrows. Each of the morphisms of schemes $g_{i, j} : W_{i, j} \to U$ is flat and they are jointly surjective. Similarly, for each fixed $i$ the morphisms of schemes $h_{i, j} : W_{i, j} \to U_ i$ are flat and jointly surjective. By Sheaves on Stacks, Lemma 95.12.2 the pullback by $(g_{i, j})_{small}$ of the restriction $\varphi |_{U_{\acute{e}tale}}$ is the restriction $\varphi |_{(W_{i, j})_{\acute{e}tale}}$ and the pullback by $(h_{i, j})_{small}$ of the restriction $\varphi |_{(U_ i)_{\acute{e}tale}}$ is the restriction $\varphi |_{(W_{i, j})_{\acute{e}tale}}$. Pullback of quasi-coherent modules by a flat morphism of schemes is exact and pullback by a jointly surjective family of flat morphisms of schemes reflects injective, resp. surjective, resp. bijective maps of quasi-coherent modules (in fact this holds for all modules as we can check exactness at stalks). Thus we see

\[ \varphi |_{U_{\acute{e}tale}} \text{ injective} \Leftrightarrow \varphi |_{(W_{i, j})_{\acute{e}tale}} \text{ injective for all }i, j \Leftrightarrow \varphi |_{(U_ i)_{\acute{e}tale}} \text{ injective for all }i \]

This finishes the proof. $\square$

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