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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

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

Comment #2651 by on

Nope because you can have things like where 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 on

In comment #2651 I forgot to say: is the open immersion and is extension by zero.

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  • 2 comment(s) on Section 30.9: Coherent sheaves on locally Noetherian schemes

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