103.4 Pullback of quasi-coherent modules

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. It is a very general fact that quasi-coherent modules on ringed topoi are compatible with pullbacks. In particular the pullback $f^*$ preserves quasi-coherent modules and we obtain a functor

$f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}),$

see Sheaves on Stacks, Lemma 96.11.2. In general this functor isn't exact, but if $f$ is flat then it is.

Lemma 103.4.1. If $f : \mathcal{X} \to \mathcal{Y}$ is a flat morphism of algebraic stacks then $f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is an exact functor.

Proof. Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose a scheme $U$ and a surjective smooth morphism $U \to V \times _\mathcal {Y} \mathcal{X}$. Then $U \to \mathcal{X}$ is still smooth and surjective as a composition of two such morphisms. From the commutative diagram

$\xymatrix{ U \ar[d] \ar[r]_{f'} & V \ar[d] \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} }$

we obtain a commutative diagram

$\xymatrix{ \mathit{QCoh}(\mathcal{O}_ U) & \mathit{QCoh}(\mathcal{O}_ V) \ar[l] \\ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar[u] & \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \ar[l] \ar[u] }$

of abelian categories. Our proof that the bottom two categories in this diagram are abelian showed that the vertical functors are faithful exact functors (see proof of Sheaves on Stacks, Lemma 96.15.1). Since $f'$ is a flat morphism of schemes (by our definition of flat morphisms of algebraic stacks) we see that $(f')^*$ is an exact functor on quasi-coherent sheaves on $V$. Thus we win. $\square$

Lemma 103.4.2. 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 96.15. 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 78.12.6.

For each $i$ we choose an étale 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 96.12.2 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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