A reference for this section is [FAC].
The category of coherent sheaves on a ringed space $X$ is a more reasonable object than the category of quasi-coherent sheaves, in the sense that it is at least an abelian subcategory of $\textit{Mod}(\mathcal{O}_ X)$ no matter what $X$ is. On the other hand, the pullback of a coherent module is “almost never” coherent in the general setting of ringed spaces.
Definition 17.12.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module if the following two conditions hold:
$\mathcal{F}$ is of finite type, and
for every open $U \subset X$ and every finite collection $s_ i \in \mathcal{F}(U)$, $i = 1, \ldots , n$ the kernel of the associated map $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type.
The category of coherent $\mathcal{O}_ X$-modules is denoted $\textit{Coh}(\mathcal{O}_ X)$.
Lemma 17.12.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Any coherent $\mathcal{O}_ X$-module is of finite presentation and hence quasi-coherent.
Proof. Let $\mathcal{F}$ be a coherent sheaf on $X$. Pick a point $x \in X$. By (1) of the definition of coherent, we may find an open neighbourhood $U$ and sections $s_ i$, $i = 1, \ldots , n$ of $\mathcal{F}$ over $U$ such that $\Psi : \bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}$ is surjective. By (2) of the definition of coherent, we may find an open neighbourhood $V$, $x \in V \subset U$ and sections $t_1, \ldots , t_ m$ of $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ V$ which generate the kernel of $\Psi |_ V$. Then over $V$ we get the presentation
as desired. $\square$
Example 17.12.3. Suppose that $X$ is a point. In this case the definition above gives a notion for modules over rings. What does the definition of coherent mean? It is closely related to the notion of Noetherian, but it is not the same: Namely, the ring $R = \mathbf{C}[x_1, x_2, x_3, \ldots ]$ is coherent as a module over itself but not Noetherian as a module over itself. See Algebra, Section 10.90 for more discussion.
Lemma 17.12.4. Let $(X, \mathcal{O}_ X)$ be a ringed space.
Any finite type subsheaf of a coherent sheaf is coherent.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism from a finite type sheaf $\mathcal{F}$ to a coherent sheaf $\mathcal{G}$. Then $\mathop{\mathrm{Ker}}(\varphi )$ is of finite type.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of coherent $\mathcal{O}_ X$-modules. Then $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are coherent.
Given a short exact sequence of $\mathcal{O}_ X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ if two out of three are coherent so is the third.
The category $\textit{Coh}(\mathcal{O}_ X)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ X)$. In particular, the category of coherent modules is abelian and the inclusion functor $\textit{Coh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ is exact.
Proof. Condition (2) of Definition 17.12.1 holds for any subsheaf of a coherent sheaf. Thus we get (1).
Assume the hypotheses of (2). Let us show that $\mathop{\mathrm{Ker}}(\varphi )$ is of finite type. Pick $x \in X$. Choose an open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{F}|_ U$ is generated by $s_1, \ldots , s_ n$. By Definition 17.12.1 the kernel $\mathcal{K}$ of the induced map $\bigoplus _{i = 1}^ n \mathcal{O}_ U \to \mathcal{G}$, $e_ i \mapsto \varphi (s_ i)$ is of finite type. Hence $\mathop{\mathrm{Ker}}(\varphi )$ which is the image of the composition $\mathcal{K} \to \bigoplus _{i = 1}^ n \mathcal{O}_ U \to \mathcal{F}$ is of finite type.
Assume the hypotheses of (3). By (2) the kernel of $\varphi $ is of finite type and hence by (1) it is coherent.
With the same hypotheses let us show that $\mathop{\mathrm{Coker}}(\varphi )$ is coherent. Since $\mathcal{G}$ is of finite type so is $\mathop{\mathrm{Coker}}(\varphi )$. Let $U \subset X$ be open and let $\overline{s}_ i \in \mathop{\mathrm{Coker}}(\varphi )(U)$, $i = 1, \ldots , n$ be sections. We have to show that the kernel of the associated morphism $\overline{\Psi } : \bigoplus _{i = 1}^ n \mathcal{O}_ U \to \mathop{\mathrm{Coker}}(\varphi )$ is of finite type. There exists an open covering of $U$ such that on each open all the sections $\overline{s}_ i$ lift to sections $s_ i$ of $\mathcal{G}$. Hence we may assume this is the case over $U$. We may in addition assume there are sections $t_ j$, $j = 1, \ldots , m$ of $\mathop{\mathrm{Im}}(\varphi )$ over $U$ which generate $\mathop{\mathrm{Im}}(\varphi )$ over $U$. Let $\Phi : \bigoplus _{j = 1}^ m \mathcal{O}_ U \to \mathop{\mathrm{Im}}(\varphi )$ be defined using $t_ j$ and $\Psi : \bigoplus _{j = 1}^ m \mathcal{O}_ U \oplus \bigoplus _{i = 1}^ n \mathcal{O}_ U \to \mathcal{G}$ using $t_ j$ and $s_ i$. Consider the following commutative diagram
By the snake lemma we get an exact sequence $\mathop{\mathrm{Ker}}(\Psi ) \to \mathop{\mathrm{Ker}}(\overline{\Psi }) \to 0$. Since $\mathop{\mathrm{Ker}}(\Psi )$ is a finite type module, we see that $\mathop{\mathrm{Ker}}(\overline{\Psi })$ has finite type.
Proof of part (4). Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\mathcal{O}_ X$-modules. By part (3) it suffices to prove that if $\mathcal{F}_1$ and $\mathcal{F}_3$ are coherent so is $\mathcal{F}_2$. By Lemma 17.9.3 we see that $\mathcal{F}_2$ has finite type. Let $s_1, \ldots , s_ n$ be finitely many local sections of $\mathcal{F}_2$ defined over a common open $U$ of $X$. We have to show that the module of relations $\mathcal{K}$ between them is of finite type. Consider the following commutative diagram
with obvious notation. By the snake lemma we get a short exact sequence $0 \to \mathcal{K} \to \mathcal{K}_3 \to \mathcal{F}_1$ where $\mathcal{K}_3$ is the module of relations among the images of the sections $s_ i$ in $\mathcal{F}_3$. Since $\mathcal{F}_1$ is coherent we see that $\mathcal{K}$ is the kernel of a map from a finite type module to a coherent module and hence finite type by (2).
Proof of (5). This follows because (3) and (4) show that Homology, Lemma 12.10.3 applies. $\square$
Lemma 17.12.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume $\mathcal{O}_ X$ is a coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is coherent if and only if it is of finite presentation.
Proof. Omitted. $\square$
Lemma 17.12.6. Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $x \in X$. Assume $\mathcal{G}$ of finite type, $\mathcal{F}$ coherent and the map on stalks $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ injective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi |_ U$ is injective.
Proof. Denote $\mathcal{K} \subset \mathcal{G}$ the kernel of $\varphi $. By Lemma 17.12.4 we see that $\mathcal{K}$ is a finite type $\mathcal{O}_ X$-module. Our assumption is that $\mathcal{K}_ x = 0$. By Lemma 17.9.5 there exists an open neighbourhood $U$ of $x$ such that $\mathcal{K}|_ U = 0$. Then $U$ works. $\square$
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