The Stacks project

Lemma 102.13.5. Let $\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack. Let $I$ be a directed set and let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system over $I$ of $\mathcal{O}_\mathcal {X}$-modules. Let $\mathcal{G}$ be an $\mathcal{O}_\mathcal {X}$-module of finite presentation. Then we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathcal{F}_ i) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i). \]

In particular, $\mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, -)$ commutes with filtered colimits in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. The displayed equality is a special case of Modules on Sites, Lemma 18.27.12. In order to apply it, we need to check the hypotheses of Sites, Lemma 7.17.8 part (4) for the site $\mathcal{X}_{fppf}$. In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 7.17.9. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{fppf})$ be the set of objects lying over affine schemes. In other words, an element of $\mathcal{B}$ is a morphism $x : U \to \mathcal{X}$ with $U$ affine. We check each of the conditions (2)(a), (2)(b), and (2)(c) of the remark in turn:

  1. Since $\mathcal{X}$ is quasi-compact, there exists a surjetive and smooth morphism $x : U \to \mathcal{X}$ with $U$ affine (Properties of Stacks, Lemma 99.6.2). Then $h_ x^\# \to *$ is a surjective map of sheaves on $\mathcal{X}_{fppf}$.

  2. Since coverings in $\mathcal{X}_{fppf}$ are fppf coverings, we see that every covering of $U \in \mathcal{B}$ is refined by a finite affine fppf covering, see Topologies, Lemma 34.7.4.

  3. Let $x : U \to \mathcal{X}$ and $x' : U' \to \mathcal{X}$ be in $\mathcal{B}$. The product $h_ x^\# \times h_{x'}^\# $ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is equal to the sheaf on $\mathcal{X}_{fppf}$ determined by the algebraic space $W = U \times _{x, \mathcal{X}, x'} U'$ over $\mathcal{X}$: for an object $y : V \to \mathcal{X}$ of $\mathcal{X}_{fppf}$ we have $(h_ x^\# \times h_{x'}^\# )(y) = \{ f : V \to W \mid y = x \circ \text{pr}_1 \circ f = x' \circ \text{pr}_2 \circ f\} $. The algebraic space $W$ is quasi-compact because $\mathcal{X}$ is quasi-separated, see Morphisms of Stacks, Lemma 100.7.8 for example. Hence we can choose an affine scheme $U''$ and a surjective ├ętale morphism $U'' \to W$. Denote $x'' : U'' \to \mathcal{X}$ the composition of $U'' \to W$ and $W \to \mathcal{X}$. Then $h_{x''}^\# \to h_ x^\# \times h_{x'}^\# $ is surjective as desired.

For the final statement, observe that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ commutes with colimits and that finitely presented modules are quasi-coherent. See Sheaves on Stacks, Lemma 95.15.1. $\square$


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