The Stacks project

Lemma 103.17.1. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module. The following are equivalent

  1. $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_\mathcal {X}$-module,

  2. $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module of finite presentation,

  3. $\mathcal{F}$ is quasi-coherent and for any morphism $f : U \to \mathcal{X}$ where $U$ is a locally Noetherian algebraic space, the pullback $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is coherent, and

  4. $\mathcal{F}$ is quasi-coherent and there exists an algebraic space $U$ and a morphism $f : U \to \mathcal{X}$ which is locally of finite type, flat, and surjective, such that the pullback $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is coherent.

Proof. Let $f : U \to \mathcal{X}$ be as in (4). Then $U$ is locally Noetherian (Morphisms of Stacks, Lemma 101.17.5) and we see that the statement of the lemma makes sense. Additionally, $f$ is locally of finite presentation by Morphisms of Stacks, Lemma 101.27.5. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $V$. In order to prove (2) we have to show that, after replacing $V$ by the members of an fppf covering of $V$, the restriction $x^*\mathcal{F}$ has a global finite presentation on $\mathcal{X}/x \cong (\mathit{Sch}/V)_{fppf}$. The projection $W = U \times _\mathcal {X} V \to V$ is locally of finite presentation, flat, and surjective. Hence we may replace $V$ by the members of an étale covering of $W$ by schemes and assume we have a morphism $h : V \to U$ with $f \circ h = x$. Since $\mathcal{F}$ is quasi-coherent, we see that the restriction $x^*\mathcal{F}$ is the pullback of $h_{small}^*(f^*\mathcal{F})|_{U_{\acute{e}tale}}$ by $\pi _ V$, see Sheaves on Stacks, Lemma 96.14.2. Since $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ locally in the étale topology has a finite presentation by assumption, we conclude (4) $\Rightarrow $ (2).

Part (2) implies (1) for any ringed topos (immediate from the definition). The properties “finite type” and “quasi-coherent” are preserved under pullback by any morphism of ringed topoi, see Modules on Sites, Lemma 18.23.4. Hence (1) implies (3), see Cohomology of Spaces, Lemma 69.12.2. Finally, (3) trivially implies (4). $\square$

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