Definition 69.12.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. A quasi-coherent module $\mathcal{F}$ on $X$ is called coherent if $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module on the site $X_{\acute{e}tale}$ in the sense of Modules on Sites, Definition 18.23.1.
69.12 Coherent modules on locally Noetherian algebraic spaces
This section is the analogue of Cohomology of Schemes, Section 30.9. In Modules on Sites, Definition 18.23.1 we have defined coherent modules on any ringed topos. We use this notion to define coherent modules on locally Noetherian algebraic spaces. Although it is possible to work with coherent modules more generally we resist the urge to do so.
This definition is compatible with the already existing notion of a coherent module on a locally Noetherian scheme; see assertion (5) of Properties of Spaces, Section 66.30 (or more directly Descent, Lemma 35.8.10). Thus from now on, if $X$ is a locally Noetherian scheme over $S$, we will not distinguish between a coherent module on $X$ viewed as a scheme or a coherent module on $X$ viewed as an algebraic space; this is compatible with the corresponding identifications of categories of quasi-coherent modules discussed in Properties of Spaces, Section 66.29.
Having said the above, the following lemma gives an understandable characterization of coherent modules on locally Noetherian algebraic spaces.
Lemma 69.12.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. The following are equivalent
$\mathcal{F}$ is coherent,
$\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_ X$-module,
$\mathcal{F}$ is a finitely presented $\mathcal{O}_ X$-module,
for any étale morphism $\varphi : U \to X$ where $U$ is a scheme the pullback $\varphi ^*\mathcal{F}$ is a coherent module on $U$, and
there exists a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that the pullback $\varphi ^*\mathcal{F}$ is a coherent module on $U$.
In particular $\mathcal{O}_ X$ is coherent, any invertible $\mathcal{O}_ X$-module is coherent, and more generally any finite locally free $\mathcal{O}_ X$-module is coherent.
Proof. To be sure, if $X$ is a locally Noetherian algebraic space and $U \to X$ is an étale morphism, then $U$ is locally Noetherian, see Properties of Spaces, Section 66.7. The lemma then follows from the points (1) – (5) made in Properties of Spaces, Section 66.30 and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma 30.9.1. $\square$
Lemma 69.12.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The category of coherent $\mathcal{O}_ X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_ X$-modules are coherent. Any extension of coherent sheaves is coherent.
Proof. Choose a scheme $U$ and a surjective étale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (66.26.1.1). By Lemma 69.12.2 we can check whether an $\mathcal{O}_ X$-module $\mathcal{F}$ is coherent by checking whether $f^*\mathcal{F}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 30.9.2. $\square$
Coherent modules form a Serre subcategory of the category of quasi-coherent $\mathcal{O}_ X$-modules. This does not hold for modules on a general ringed topos.
Lemma 69.12.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.
Proof. Choose a scheme $U$ and a surjective étale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (66.26.1.1). By Lemma 69.12.2 we can check whether an $\mathcal{O}_ X$-module $\mathcal{G}$ is coherent by checking whether $f^*\mathcal{H}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 30.9.3. $\square$
Lemma 69.12.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$,. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-modules $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ are coherent.
Proof. Via Lemma 69.12.2 this follows from the result for schemes, see Cohomology of Schemes, Lemma 30.9.4. $\square$
Lemma 69.12.6. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$.
If $\mathcal{F}_{\overline{x}} = 0$ then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\mathcal{F}|_{X'} = 0$.
If $\varphi _{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is injective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi |_{X'}$ is injective.
If $\varphi _{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is surjective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi |_{X'}$ is surjective.
If $\varphi _{\overline{x}} : \mathcal{G}_{\overline{x}} \to \mathcal{F}_{\overline{x}}$ is bijective, then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\varphi |_{X'}$ is an isomorphism.
Proof. Let $\varphi : U \to X$ be an étale morphism where $U$ is a scheme and let $u \in U$ be a point mapping to $x$. By Properties of Spaces, Lemmas 66.29.4 and 66.22.1 as well as More on Algebra, Lemma 15.45.1 we see that $\varphi _{\overline{x}}$ is injective, surjective, or bijective if and only if $\varphi _ u : \varphi ^*\mathcal{F}_ u \to \varphi ^*\mathcal{G}_ u$ has the corresponding property. Thus we can apply the schemes version of this lemma to see that (after possibly shrinking $U$) the map $\varphi ^*\mathcal{F} \to \varphi ^*\mathcal{G}$ is injective, surjective, or an isomorphism. Let $X' \subset X$ be the open subspace corresponding to $|\varphi |(|U|) \subset |X|$, see Properties of Spaces, Lemma 66.4.8. Since $\{ U \to X'\} $ is a covering for the étale topology, we conclude that $\varphi |_{X'}$ is injective, surjective, or an isomorphism as desired. Finally, observe that (1) follows from (2) by looking at the map $\mathcal{F} \to 0$. $\square$
Lemma 69.12.7. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and $\mathcal{G}$ the quasi-coherent $\mathcal{O}_ Z$-module such that $i_*\mathcal{G} = \mathcal{F}$, see Morphisms of Spaces, Definition 67.15.4. Then $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module.
Proof. The statement of the lemma makes sense as a coherent module is in particular of finite type. Moreover, as $Z \to X$ is a closed immersion it is locally of finite type and hence $Z$ is locally Noetherian, see Morphisms of Spaces, Lemmas 67.23.7 and 67.23.5. Finally, as $\mathcal{G}$ is of finite type it is a coherent $\mathcal{O}_ Z$-module by Lemma 69.12.2 $\square$
Lemma 69.12.8. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally Noetherian algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $i_*$ induces an equivalence between the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{I}$ and the category of coherent $\mathcal{O}_ Z$-modules.
Proof. The functor is fully faithful by Morphisms of Spaces, Lemma 67.14.1. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}$. By Morphisms of Spaces, Lemma 67.14.1 we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. To check that $\mathcal{G}$ is coherent we can work étale locally (Lemma 69.12.2). Choosing an étale covering by a scheme we conclude that $\mathcal{G}$ is coherent by the case of schemes (Cohomology of Schemes, Lemma 30.9.8). Hence the functor is fully faithful and the proof is done. $\square$
Lemma 69.12.9. Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$ with $Y$ locally Noetherian. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^ pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F}$ is coherent.
Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Then $V \times _ Y X \to V$ is a finite morphism of locally Noetherian schemes. By (69.3.0.1) we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.9.9. $\square$
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