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Chapter 29: Cohomology of Schemes > Section 29.9: 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. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 2025–2043 (see updates for more information).

    \begin{lemma}
    \label{lemma-coherent-Noetherian}
    Let $X$ be a locally Noetherian scheme.
    Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{F}$ is coherent,
    \item $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module,
    \item $\mathcal{F}$ is a finitely presented $\mathcal{O}_X$-module,
    \item for any affine open $\Spec(A) = U \subset X$ we have
    $\mathcal{F}|_U = \widetilde M$ with $M$ a finite $A$-module, and
    \item there exists an affine open covering $X = \bigcup U_i$,
    $U_i = \Spec(A_i)$ such that each
    $\mathcal{F}|_{U_i} = \widetilde M_i$ with $M_i$ a finite $A_i$-module.
    \end{enumerate}
    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.
    \end{lemma}
    
    \begin{proof}
    The implications (1) $\Rightarrow$ (2) and (1) $\Rightarrow$ (3) hold
    in general, see
    Modules, Lemma \ref{modules-lemma-coherent-finite-presentation}.
    If $\mathcal{F}$ is finitely presented then $\mathcal{F}$ is
    quasi-coherent, see
    Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-coherent}.
    Hence also (3) $\Rightarrow$ (2).
    
    \medskip\noindent
    Assume $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module.
    By
    Properties, Lemma \ref{properties-lemma-finite-type-module}
    we see that on any affine open
    $\Spec(A) = U \subset X$ 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 \ref{properties-lemma-finite-presentation-module}.
    In other words (2) $\Rightarrow$ (3).
    
    \medskip\noindent
    By Modules, Lemma \ref{modules-lemma-coherent-structure-sheaf} 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 $U = \Spec(A)$ is affine. In this case
    $f_1, \ldots, f_n \in A$ are elements of $A$. Since $A$ is
    Noetherian, see
    Properties, Lemma \ref{properties-lemma-locally-Noetherian}
    the kernel $K$ of the map $\bigoplus_{i = 1, \ldots, n} A \to A$
    is a finite $A$-module. See for example
    Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}.
    As the functor\ $\widetilde{ }$\ is exact, see
    Schemes, Lemma \ref{schemes-lemma-spec-sheaves}
    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 \ref{properties-lemma-finite-type-module}
    again we see that $\widetilde K$ is of finite type. We conclude
    that (1), (2) and (3) are all equivalent.
    
    \medskip\noindent
    It follows from
    Properties, Lemma \ref{properties-lemma-finite-type-module}
    that (2) implies (4). It is trivial that (4) implies (5).
    The discussion in
    Schemes, Section \ref{schemes-section-quasi-coherent}
    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.
    \end{proof}

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