Lemma 30.9.3 (01Y1)—The Stacks project

Lemma 30.9.3. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$ -module. Any quasi-coherent submodule of $\mathcal{F}$ is coherent. Any quasi-coherent quotient module of $\mathcal{F}$ is coherent.

Proof. We may assume that $X$ is affine, say $X = \mathop{\mathrm{Spec}}(A)$ . Properties, Lemma 28.5.2 implies that $A$ is Noetherian. Lemma 30.9.1 turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$ -module is a finite $A$ -module, see for example Algebra, Lemma 10.51.1. $\square$

Comments (3)

Comment #2650 by Student on July 19, 2017 at 17:33

Isn't the quasi-coherence assumption superfluous? That is, any submodule of $\mathcal{F}$ is coherent, as the notions of finite type and finite presentations coincide over locally Noetherian schemes.

Comment #2651 by Johan on July 19, 2017 at 18:14

Nope because you can have things like $j_!(\mathcal{F}|_U)$ where $U \subset X$ is an open. Such modules aren't even locally generated by sections, and a fortiori not quasi-coherent and a fortiori not coherent.

Comment #2652 by Johan on July 19, 2017 at 18:15

In comment #2651 I forgot to say: $j : U \to X$ is the open immersion and $j_!$ is extension by zero.

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