29.9 Coherent sheaves on locally Noetherian schemes

We have defined the notion of a coherent module on any ringed space in Modules, Section 17.12. Although it is possible to consider coherent sheaves on non-Noetherian schemes we will always assume the base scheme is locally Noetherian when we consider coherent sheaves. Here is a characterization of coherent sheaves on locally Noetherian schemes.

Lemma 29.9.1. Let $X$ be a locally Noetherian scheme. 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 affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ we have $\mathcal{F}|_ U = \widetilde M$ with $M$ a finite $A$-module, and

5. there exists an affine open covering $X = \bigcup U_ i$, $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ such that each $\mathcal{F}|_{U_ i} = \widetilde M_ i$ with $M_ i$ a finite $A_ i$-module.

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. The implications (1) $\Rightarrow$ (2) and (1) $\Rightarrow$ (3) hold in general, see Modules, Lemma 17.12.2. If $\mathcal{F}$ is finitely presented then $\mathcal{F}$ is quasi-coherent, see Modules, Lemma 17.11.2. Hence also (3) $\Rightarrow$ (2).

Assume $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_ X$-module. By Properties, Lemma 27.16.1 we see that on any affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ we have $\mathcal{F}|_ U = \widetilde M$ with $M$ a finite $A$-module. Since $A$ is Noetherian we see that $M$ has a finite resolution

$A^{\oplus m} \to A^{\oplus n} \to M \to 0.$

Hence $\mathcal{F}$ is of finite presentation by Properties, Lemma 27.16.2. In other words (2) $\Rightarrow$ (3).

By Modules, Lemma 17.12.5 it suffices to show that $\mathcal{O}_ X$ is coherent in order to show that (3) implies (1). Thus we have to show: given any open $U \subset X$ and any finite collection of sections $f_ i \in \mathcal{O}_ X(U)$, $i = 1, \ldots , n$ the kernel of the map $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{O}_ U$ is of finite type. Since being of finite type is a local property it suffices to check this in a neighbourhood of any $x \in U$. Thus we may assume $U = \mathop{\mathrm{Spec}}(A)$ is affine. In this case $f_1, \ldots , f_ n \in A$ are elements of $A$. Since $A$ is Noetherian, see Properties, Lemma 27.5.2 the kernel $K$ of the map $\bigoplus _{i = 1, \ldots , n} A \to A$ is a finite $A$-module. See for example Algebra, Lemma 10.50.1. As the functor $\widetilde{ }$ is exact, see Schemes, Lemma 25.5.4 we get an exact sequence

$\widetilde K \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{O}_ U$

and by Properties, Lemma 27.16.1 again we see that $\widetilde K$ is of finite type. We conclude that (1), (2) and (3) are all equivalent.

It follows from Properties, Lemma 27.16.1 that (2) implies (4). It is trivial that (4) implies (5). The discussion in Schemes, Section 25.24 show that (5) implies that $\mathcal{F}$ is quasi-coherent and it is clear that (5) implies that $\mathcal{F}$ is of finite type. Hence (5) implies (2) and we win. $\square$

Lemma 29.9.2. Let $X$ be a locally Noetherian scheme. 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. This is a restatement of Modules, Lemma 17.12.4 in a particular case. $\square$

The following lemma does not always hold for the category of coherent $\mathcal{O}_ X$-modules on a general ringed space $X$.

Lemma 29.9.3. Let $X$ be a locally Noetherian scheme. 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. We may assume that $X$ is affine, say $X = \mathop{\mathrm{Spec}}(A)$. Properties, Lemma 27.5.2 implies that $A$ is Noetherian. Lemma 29.9.1 turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$-module is a finite $A$-module, see for example Algebra, Lemma 10.50.1. $\square$

Lemma 29.9.4. Let $X$ be a locally Noetherian scheme. 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. It is shown in Modules, Lemma 17.20.5 that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent. The result for tensor products is Modules, Lemma 17.15.5 $\square$

Lemma 29.9.5. Let $X$ be a locally Noetherian scheme. 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 $x \in X$.

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

2. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is injective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is injective.

3. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is surjective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is surjective.

4. If $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ is bijective, then there exists an open neighbourhood $U \subset X$ of $x$ such that $\varphi |_ U$ is an isomorphism.

Lemma 29.9.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Let $x \in X$. Suppose $\psi : \mathcal{G}_ x \to \mathcal{F}_ x$ is a map of $\mathcal{O}_{X, x}$-modules. Then there exists an open neighbourhood $U \subset X$ of $x$ and a map $\varphi : \mathcal{G}|_ U \to \mathcal{F}|_ U$ such that $\varphi _ x = \psi$.

Proof. In view of Lemma 29.9.1 this is a reformulation of Modules, Lemma 17.20.3. $\square$

Lemma 29.9.7. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition 28.5.5.

Proof. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$ such that $i_*\mathcal{G} \cong \mathcal{F}$. Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed. The scheme $Z$ is locally Noetherian by Morphisms, Lemmas 28.14.5 and 28.14.6. Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module by Lemma 29.9.1 $\square$

Lemma 29.9.8. Let $i : Z \to X$ be a closed immersion of locally Noetherian schemes. 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, Lemma 28.4.1. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}$. By Morphisms, Lemma 28.4.1 we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. By Modules, Lemma 17.13.3 we see that $\mathcal{G}$ is of finite type. Hence $\mathcal{G}$ is coherent by Lemma 29.9.1. Thus the functor is also essentially surjective as desired. $\square$

Lemma 29.9.9. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-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 if $\mathcal{F}$ is coherent.

Proof. The higher direct images vanish by Lemma 29.2.3 and because a finite morphism is affine (by definition). Note that the assumptions imply that also $X$ is locally Noetherian (see Morphisms, Lemma 28.14.6) and hence the statement makes sense. Let $\mathop{\mathrm{Spec}}(A) = V \subset Y$ be an affine open subset. By Morphisms, Definition 28.42.1 we see that $f^{-1}(V) = \mathop{\mathrm{Spec}}(B)$ with $A \to B$ finite. Lemma 29.9.1 turns the statement of the lemma into the following algebra fact: If $M$ is a finite $B$-module, then $M$ is also finite viewed as a $A$-module, see Algebra, Lemma 10.7.2. $\square$

In the situation of the lemma also the higher direct images are coherent since they vanish. We will show that this is always the case for a proper morphism between locally Noetherian schemes (insert future reference here).

Lemma 29.9.10. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq 0$. Then $\mathcal{F}$ is generated by global sections and $H^ i(X, \mathcal{F}) = 0$ for $i > 0$.

Proof. By Lemma 29.9.7 we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module. Since the dimension of $Z$ is $0$, we see $Z$ is a disjoint union of affines (Properties, Lemma 27.10.5). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Lemma 29.2.2). Hence $\mathcal{F} = i_*\mathcal{G}$ is globally generated. Since the cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree (Lemma 29.2.4 applies as a closed immersion is affine) we conclude that the higher cohomology groups of $\mathcal{F}$ are zero. $\square$

Lemma 29.9.11. Let $X$ be a scheme. Let $j : U \to X$ be the inclusion of an open. Let $T \subset X$ be a closed subset contained in $U$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ U$-module with $\text{Supp}(\mathcal{F}) \subset T$, then $j_*\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module.

Proof. Consider the open covering $X = U \cup (X \setminus T)$. Then $j_*\mathcal{F}|_ U = \mathcal{F}$ is coherent and $j_*\mathcal{F}|_{X \setminus T} = 0$ is also coherent. Hence $j_*\mathcal{F}$ is coherent. $\square$

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