The Stacks project

Lemma 30.9.1. 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 28.16.1 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 28.16.2. 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 28.5.2 the kernel $K$ of the map $\bigoplus _{i = 1, \ldots , n} A \to A$ is a finite $A$-module. See for example Algebra, Lemma 10.51.1. As the functor $\widetilde{ }$ is exact, see Schemes, Lemma 26.5.4 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 28.16.1 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 28.16.1 that (2) implies (4). It is trivial that (4) implies (5). The discussion in Schemes, Section 26.24 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. $\square$

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