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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.

  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 $\text{Ker}(\varphi)$ is finite type.
  3. 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.
  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 $\text{Ker}(\varphi)$ 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}

    Comments (6)

    Comment #975 by JuanPablo on September 2, 2014 a 10:25 pm UTC

    To prove that the category of coherent modules is abelian we need that the sum of two coherent modules is coherent, I did not find that fact immediate so I put here the proof I have:

    Suppose $\mathcal{F}$ and $\mathcal{G}$ coherent, then $\mathcal{F} \oplus \mathcal{G}$ is of finite type. Now take $\bigoplus_{i=1}^n \mathcal{O}_U \rightarrow \mathcal{F}|_U \oplus \mathcal{G}|_U$, which corresponds to a pair of morphisms $\bigoplus_{i=1}^n \mathcal{O}_U \rightarrow \mathcal{F}|_U$ and $\bigoplus_{i=1}^n \mathcal{O}_U \rightarrow \mathcal{G}|_U$, with kernels $K_1$ and $K_2$. The kernel of the original map is $K_1\cap K_2$. Now we have the exact sequence: $0\rightarrow K_1 \cap K_2\rightarrow K_1\oplus K_2\rightarrow \bigoplus_{i=1}^n\mathcal{O}_U$ where the arrow in the right is the difference. Then $K_1\cap K_2$ is of finite type because $K_1\oplus K_2$ is of finite type and $\bigoplus_{i=1}^n\mathcal{O}_U$ is of finite presentation (tag 01BP (2)).

    Comment #976 by JuanPablo on September 2, 2014 a 10:39 pm UTC

    No, wait;

    $K_1\oplus K_2\rightarrow \bigoplus \mathcal{O}_U$ is not surjective so it does not work. Now I don't know how to prove that the sum of two coherent modules is coherent.

    Comment #977 by JuanPablo on September 2, 2014 a 10:56 pm UTC

    Ok. I think is as follows:

    The kernel of $K_1\rightarrow \bigoplus_{i=1}^n \mathcal{O}_U \rightarrow \mathcal{G}|_U$ is $K_1\cap K_2$ which is of finite type because of (2) in this lemma.

    Comment #1008 by Johan (site) on September 6, 2014 a 5:20 pm UTC

    Hi! In stead of directly arguing this in this case, I have put in a reference to Lemma 0754 which says that in the situation where you have a full subcat preserved under kernels and cokernels and extensions, you always have an abelian subcategory. Hope this clarifies things. The change is here.

    Comment #2366 by Katharina on February 7, 2017 a 8:30 am UTC

    In the proof of part (3) it is implicitly assumed that $$ \oplus_{i=1}^n \mathcal{O}_U \to \mathcal{G} $$ is surjective, which is not the case in general.

    Comment #2429 by Johan (site) on February 17, 2017 a 2:35 pm UTC

    Sorry, I do not understand your question. Can you clarify? Thanks.

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