## Tag `01BY`

Chapter 17: Sheaves of Modules > Section 17.12: Coherent modules

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 $\text{Ker}(\varphi)$ is finite type.
- Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of coherent $\mathcal{O}_X$-modules. Then $\text{Ker}(\varphi)$ and $\text{Coker}(\varphi)$ 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 $\text{Ker}(\varphi)$ 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 $\text{Ker}(\varphi)$ 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 $\text{Coker}(\varphi)$ is coherent. Since $\mathcal{G}$ is of finite type so is $\text{Coker}(\varphi)$. Let $U \subset X$ be open and let $\overline{s}_i \in \text{Coker}(\varphi)(U)$, $i = 1, \ldots, n$ be sections. We have to show that the kernel of the associated morphism $\overline{\Psi} : \bigoplus_{i = 1}^n \mathcal{O}_U \to \text{Coker}(\varphi)$ has 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 $U$. Thus $\overline{\Psi}$ lifts to $\Psi : \bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{G}$ Consider the following diagram $$ \xymatrix{ 0 \ar[r] & \text{Ker}(\Psi) \ar[r] \ar[d] & \bigoplus_{i = 1}^n \mathcal{O}_U \ar[r] \ar@{=}[d] & \mathcal{G} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \text{Ker}(\overline{\Psi}) \ar[r] & \bigoplus_{i = 1}^n \mathcal{O}_U \ar[r] & \text{Coker}(\varphi) \ar[r] & 0 } $$ By the snake lemma we get a short exact sequence $0 \to \text{Ker}(\Psi) \to \text{Ker}(\overline{\Psi}) \to \text{Im}(\varphi) \to 0$. Hence by Lemma 17.9.3 we see that $\text{Ker}(\overline{\Psi})$ 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 $\mathcal{F}_3$ is coherent we see that $\mathcal{K}_3$ is finite type. Since $\mathcal{F}_1$ is coherent we see that the image $\mathcal{I}$ of $\mathcal{K}_3 \to \mathcal{F}_1$ is coherent. Hence $\mathcal{K}$ is the kernel of the map $\mathcal{K}_3 \to \mathcal{I}$ between a finite type sheaf and a coherent sheaves and hence finite type by (2).

Proof of (5). This follows because (3) and (4) show that Homology, Lemma 12.9.3 applies. $\square$

The code snippet corresponding to this tag is a part of the file `modules.tex` and is located in lines 1691–1712 (see updates for more information).

```
\begin{lemma}
\label{lemma-coherent-abelian}
Let $(X, \mathcal{O}_X)$ be a ringed space.
\begin{enumerate}
\item Any finite type subsheaf of a coherent sheaf is coherent.
\item Let $\varphi : \mathcal{F} \to \mathcal{G}$
be a morphism from a finite type sheaf $\mathcal{F}$
to a coherent sheaf $\mathcal{G}$. Then $\Ker(\varphi)$ is finite type.
\item Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism
of coherent $\mathcal{O}_X$-modules. Then
$\Ker(\varphi)$ and
$\Coker(\varphi)$ are coherent.
\item 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.
\item 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.
\end{enumerate}
\end{lemma}
\begin{proof}
Condition (2) of Definition \ref{definition-coherent}
holds for any subsheaf of a coherent sheaf. Thus we get (1).
\medskip\noindent
Assume the hypotheses of (2).
Let us show that $\Ker(\varphi)$ 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 \ref{definition-coherent} 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 $\Ker(\varphi)$ which is the image of the
composition
$\mathcal{K} \to \bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{F}$
is of finite type.
\medskip\noindent
Assume the hypotheses of (3).
By (2) the kernel of $\varphi$ is of finite type and
hence by (1) it is coherent.
\medskip\noindent
With the same hypotheses
let us show that $\Coker(\varphi)$ is coherent.
Since $\mathcal{G}$ is of finite type so is $\Coker(\varphi)$.
Let $U \subset X$ be open and let
$\overline{s}_i \in \Coker(\varphi)(U)$,
$i = 1, \ldots, n$ be sections. We have to show that
the kernel of the associated morphism
$\overline{\Psi} : \bigoplus_{i = 1}^n \mathcal{O}_U \to \Coker(\varphi)$
has 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 $U$. Thus $\overline{\Psi}$ lifts to
$\Psi : \bigoplus_{i = 1}^n \mathcal{O}_U \to \mathcal{G}$
Consider the following diagram
$$
\xymatrix{
0 \ar[r] &
\Ker(\Psi) \ar[r] \ar[d] &
\bigoplus_{i = 1}^n \mathcal{O}_U \ar[r] \ar@{=}[d] &
\mathcal{G} \ar[r] \ar[d] &
0 \\
0 \ar[r] &
\Ker(\overline{\Psi}) \ar[r] &
\bigoplus_{i = 1}^n \mathcal{O}_U \ar[r] &
\Coker(\varphi) \ar[r] &
0
}
$$
By the snake lemma we get a short exact sequence
$0 \to \Ker(\Psi) \to \Ker(\overline{\Psi})
\to \Im(\varphi) \to 0$. Hence by
Lemma \ref{lemma-extension-finite-type} we
see that $\Ker(\overline{\Psi})$ has finite type.
\medskip\noindent
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 \ref{lemma-extension-finite-type} 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 $\mathcal{F}_3$ is coherent we see that
$\mathcal{K}_3$ is finite type. Since $\mathcal{F}_1$
is coherent we see that the image $\mathcal{I}$
of $\mathcal{K}_3 \to \mathcal{F}_1$
is coherent. Hence $\mathcal{K}$
is the kernel of the map $\mathcal{K}_3 \to \mathcal{I}$
between a finite type sheaf and a coherent sheaves and hence
finite type by (2).
\medskip\noindent
Proof of (5). This follows because (3) and (4) show that
Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}
applies.
\end{proof}
```

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