The Stacks project

17.12 Coherent modules

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:

  1. $\mathcal{F}$ is of finite type, and

  2. 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

\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ V \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ V \to \mathcal{F}|_ V \to 0 \]

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.

  1. Any finite type subsheaf of a coherent sheaf is coherent.

  2. 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.

  3. 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.

  4. 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.

  5. 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

\[ \xymatrix{ 0 \ar[r] & \bigoplus _{j = 1}^ m \mathcal{O}_ U \ar[d]_\Phi \ar[r] & \bigoplus _{j = 1}^ m \mathcal{O}_ U \oplus \bigoplus _{i = 1}^ n \mathcal{O}_ U \ar[d]_\Psi \ar[r] & \bigoplus _{i = 1}^ n \mathcal{O}_ U \ar[d]_{\overline{\Psi }} \ar[r] & 0 \\ 0 \ar[r] & \mathop{\mathrm{Im}}(\varphi ) \ar[r] & \mathcal{G} \ar[r] & \mathop{\mathrm{Coker}}(\varphi ) \ar[r] & 0 } \]

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

\[ \xymatrix{ 0 \ar[r] & 0 \ar[r] \ar[d] & \bigoplus _{i = 1}^{n} \mathcal{O}_ U \ar[r] \ar[d] & \bigoplus _{i = 1}^{n} \mathcal{O}_ U \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1 \ar[r] & \mathcal{F}_2 \ar[r] & \mathcal{F}_3 \ar[r] & 0 } \]

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. $\square$

Comments (4)

Comment #4738 by B. Shih on

Sorry, I don't see why it was necessary to pass to the image in proof (4) of Lemma 01BY. Do we not already have being the kenrel of ? By (2) it is of finite type.

Comment #4822 by on

Thanks and fixed here. If you want to be listed as a collaborator on the Stacks project, let me know your first name too.

Comment #5539 by minsom on

May I ask you something? In the proof of lemma 01BY (3) , how can we show that "There exists an open covering of such that on each open all the sections lift to sections of ." ? For sheaf axiom , we need the fact ' for open cover , , each element is same in '

Comment #5728 by on

@#5539: this is true because the cokernel is the sheafification of the presheaf cokernel and because there are only finitely many . When we are lifting these sections, say to the members of an open covering , say lifts to the section , we don't require and to restrict to the same element of . Then in the next step we replace by and we continue.

Hope this helps!

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01BU. Beware of the difference between the letter 'O' and the digit '0'.