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I use the following definition:
Definition
Let $(X,\mathscr O_X)$ be a locally ringed space.
An $\mathscr O_X$-module $\mathscr F$ is coherent if
(i) it is locally finitely generated.
(ii) for every $n,$ for every open $U\subset X,$ and for every $u:\mathscr O_X^n|_U\rightarrow \mathscr F|_U,$ the kernel of $u$ is locally finitely generated.

Then how can I show directly that, if $A$ is a Noetherian ring, $X=\operatorname{Spec} A,$ then $\mathscr O_X$ is a coherent $\mathscr O_X$-module?

P.S. Here is a related question.

Edit
An $\mathscr O_X$-module is locally finitely generated if, for each $x\in X,$ there exists a nbd. $U$ of $x$ such that $\mathscr O_X|_U$ is generated by a finite family of sections of $\mathscr O_X$ over $U.$

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awllower
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1 Answers1

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Let $X = \operatorname{Spec} A$, where $A$ is a Noetherian ring.

Condition (i) is clear. For condition (ii), let $U$ be an open subset of $X$, and let $n$ be an integer. Consider the kernel of a map $u$, $$u: \mathcal{O}_X^n |_U \rightarrow \mathcal{O}_X |_U.$$ We want to show that $\ker u$ is locally finitely generated. Let $x \in U$ and let $U' \subseteq U$ be an open affine neighborhood of $x$, so that $\mathcal{O}_X |_{U'} \cong \operatorname{Spec} B$, for some ring $B$. Then $B$ must be Noetherian (see Hartshorne Prop 3.2). Thus $\Gamma(U',\mathcal{O}_X |_U)] \cong B$ and $\Gamma(U',\mathcal{O}_X^n |_U)] \cong B^n$ is a finitely generated Noetherian module.

Now we want to show $(\ker u)|_{U'}$ is generated by a finite family of sections of $\mathcal{O}_X$ over $U'$.

Since $$\Gamma(U',\ker u) = \ker[\Gamma(U',\mathcal{O}_X^n |_U) \rightarrow \Gamma(U',\mathcal{O}_X |_U)]$$ is a submodule of a finitely generated Noetherian module, $\Gamma(U',\ker u)$ itself is finitely generated, and we are done.

John M
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  • But I am not sure if $ u $ is an A-module homomorphism, thus its kernel might not be a submodule of $ B^n?$ – awllower May 25 '14 at 12:18
  • Well, it is true that $u$ itself is a morphism of sheaves of modules. But the global section $u(X)$ of $u$ is an $A$-module homomorphism. More pertinently, for the affine opens $U'$, the section $u(U')$ of $u$ over $U'$ is a $\Gamma(U',\mathcal{O}_X)$-homomorphism, where $\Gamma(U',\mathcal{O}_X)$ is a Noetherian ring. – John M May 26 '14 at 00:16
  • How can $u,$ being a morphism of sheaves of modules over $U,$ have a morphism $u(X)$ over $X?$ In any case, I see now that this argument can show the coherence of any locally finitely generated quasi-coherent module, not only $\mathscr O_X.$ Thanks very much. – awllower May 26 '14 at 03:24
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    (1) I'll answer your question with a question: What is your definition for a morphism of sheaves of modules? (2) For you other comment, I would say "coherent = quasi-coherent + locally finitely presented". For a Noetherian module would that also be equivalent to "quasi-coherent + locally finitely generated", so be a little cautious with that. – John M May 26 '14 at 03:49
  • Anyways, I see that $u$ is a $B$-module homomorphism, thus its kernel is finitely generated. Also thanks for your two questions: they help me clarify some of the points. By the way, we might take $U'$ to be a basic open set, and then the assertion needed is clear. Thanks again. :) – awllower Jun 03 '14 at 02:23