65.12 Coherent modules on locally Noetherian algebraic spaces

This section is the analogue of Cohomology of Schemes, Section 29.9. 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.

Definition 65.12.1. 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.

Of course this definition is a bit hard to work with. We usually use the characterization given in the lemma below.

Lemma 65.12.2. 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

1. $\mathcal{F}$ is coherent,

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

3. $\mathcal{F}$ is a finitely presented $\mathcal{O}_ X$-module,

4. 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

5. 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 62.7. The lemma then follows from the points (1) – (5) made in Properties of Spaces, Section 62.30 and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma 29.9.1. $\square$

Lemma 65.12.3. 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 (62.26.1.1). By Lemma 65.12.2 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 29.9.2. $\square$

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 65.12.4. 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 (62.26.1.1). By Lemma 65.12.2 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 29.9.3. $\square$

Lemma 65.12.5. 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 65.12.2 this follows from the result for schemes, see Cohomology of Schemes, Lemma 29.9.4. $\square$

Lemma 65.12.6. 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 $x \in |X|$.

1. If $\mathcal{F}_{\overline{x}} = 0$ then there exists an open neighbourhood $X' \subset X$ of $x$ such that $\mathcal{F}|_{X'} = 0$.

2. 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.

3. 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.

4. 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 62.29.4 and 62.22.1 as well as More on Algebra, Lemma 15.44.1 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 62.4.8. 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 65.12.7. 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 63.15.4. 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 63.23.7 and 63.23.5. Finally, as $\mathcal{G}$ is of finite type it is a coherent $\mathcal{O}_ Z$-module by Lemma 65.12.2 $\square$

Lemma 65.12.8. 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 63.14.1. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}$. By Morphisms of Spaces, Lemma 63.14.1 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 65.12.2). Choosing an étale covering by a scheme we conclude that $\mathcal{G}$ is coherent by the case of schemes (Cohomology of Schemes, Lemma 29.9.8). Hence the functor is fully faithful and the proof is done. $\square$

Lemma 65.12.9. 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 (65.3.0.1) we reduce to the case of schemes which is Cohomology of Schemes, Lemma 29.9.9. $\square$

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