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To be precise, I want to strengthen the second part of Proposition 5.4 Chapter 2 in Hartshorne GTM 52 as follows:

Let $X$ be a sheme, then an $\mathcal{O}_X$-module $\mathscr{F}$ is coherent if and only if for every open affine subset $U=SpecA$ of $X$, there is an $A$-module $M$ such that $\mathscr{F}\mid_U\cong\widetilde{M}$, and $M$ be a finitely generated $A$-module.

By the defination of coherent sheaf (Hartshorne p.111), it only claims the existence of a cover of $X$ satisfies such property(i.e. "if" part comes by free).

If the Noetherian condition can be dropped in this proposition, it can also be dropped in Corollery 5.5, Proposition 5.5(b), 5.11(c) etc.

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I quote Hartshorne (p. 111): "Although we have just defined the notion of quasi-coherent and coherent sheaves on an arbitrary scheme, we will normally not mention coherent sheaves unless the scheme is noetherian. This is because the notion of coherence is not at all well-behaved on a nonnoetherian scheme." I think that answers the question in your title. As for your actual question, I'm afraid I don't have a clue. –  Zhen Lin Jul 21 '11 at 10:37

2 Answers 2

up vote 10 down vote accepted

Given a scheme $(X,\mathcal O_X)$ and a sheaf $\mathcal F$ of $O_X$-Modules, the following are equivalent:
a) There exists a covering $\mathcal U=(U_i)$ of $X$ by open subsets $U_i\subset X$ and $\mathcal O_{U_i}$-isomorphisms $\mathcal F|U_i \simeq \tilde M_i$ for some family of $\mathcal O(U_i)$-modules $M_i$.
b) For every affine open subset $U\subset X$ there exists an $\mathcal O(U)$-module $M$ ( namely $M=\mathcal F (U)$) and an $\mathcal O_{U}$-isomorphism $\mathcal F|U \simeq \tilde M.$

This equivalence is a theorem, proved for example in Mumford's Red Book, at the very beginning of Chapter III, in §1 (along with other equivalent characterizations). This has nothing to do with noetherian hypotheses.
And now on to coherent sheaves.

Recall that a a sheaf $\mathcal F$ of $O_X$-Modules is said to be finitely generated if for every $x\in X$ there exists an open neighbourhood $U$ of $x$ and a surjective sheaf homomorphism $\mathcal O_{U}^r \to \mathcal F|U \to 0$ for some integer $r$. The sheaf $\mathcal F$ is then said to be coherent if it is finitely generated and if for every open subset $V\subset X$ and every (not necessarily surjective !) morphism $\mathcal O_{V}^N \to \mathcal F|V$, the kernel is also finitely generated . Again, no noetherian hypothesis in sight. End of story? Not at all! The problem is that coherence is very difficult to check in general and actually for some schemes, even affine ones, the structure sheaf $O_X$ is not coherent, and in that sad case the concept coherent is essentially worthless . In particular, and this one of your questions, the equivalence of categories mentioned in Corollary (5.5) is FALSE without the noetherian hypothesis.

However all troubles evaporate if you assume that $X$ is locally noetherian. You then have the wonderful equivalence (implying of course that the structure sheaf $O_X$ is coherent)

$$\mathcal F \;\text {coherent} \stackrel {X \text {loc.noeth.}}{\iff} \mathcal F \; \text {finitely generated and quasi-coherent }$$

Edit I have tried to evade the issue, but since Li explicitly asks: Yes, Hartshorne's definition is incorrect. Here is what I mean.
The notion of coherent sheaf was introduced by Henri Cartan in the theory of holomorphic functions of several varables around 1944. In 1946 Oka proved that $\mathcal O_{\mathbb C^n}$ is coherent and this is a very difficult theorem, not following at all from Cartan's definition, the one I reproduced above.

In 1955, as is well known, Serre introduced coherent sheaves into Algebraic Geometry in his famous article Faisceaux Algébriques Cohérents and used the exact same definition as Cartan, as acknowledged in his Introduction.
Coherent sheaves were then defined in EGA for schemes and ringed spaces, always with Cartan's definition above. Ditto for the generalized analytic spaces (with nilpotents) introduced by Grauert (influenced by Grothendieck) around 1960. And that definition is also the one used in De Jong and collaborators's recent monumental online Stacks Project.

So the definition I reproduced above is the one adopted by the founders and in the foundational documents. To change it would be, in my opinion, very misleading and might for example induce one to believe that very profound theorems are trivial. Or worse, induce mistakes by inappropriately applying results from texts using the standard definition of "coherent sheaf".

Incidentally, Mumford very elegantly solves the definition problem: he only defines "coherent" in the noetherian case since he only only uses the notion in that case!

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I guess your defination of coherent sheaf is different from Hartshorne: X is a scheme, a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ is coherent if X can be covered by open affine $U_i=SpecA_i$, such that for each i there is a finitely generated $A_i$-module $M_i$ with $\mathcal{F}\mid U_i \cong \tilde{M_i}$.I am pretty sure at present time, the Noetherian condition can be dropped if using Hartshorne's defination. Vakil wrote in his note: it is common in the later literature to incorrectly define coherent as finitely generated. So whether Hartshorne's defination is the $incorrect$ one? –  Li Zhan Jul 22 '11 at 12:40
    
Dear Li, to put it bluntly: Yes, Hartshorne's definition is incorrect. I have written an edit. –  Georges Elencwajg Jul 23 '11 at 13:39
    
Thank you for the comprehensive explanation! I appreciate it! –  Li Zhan Jul 24 '11 at 3:09

This is not really an answer as much as a reference. Ravi Vakil's notes treats the notion of coherence, finite presentation, and finite generation in more general cases then just the Noetherian case. The important fact (as mentioned in Georges post above) is that all these conditions are equivalent on an affine Noetherian neighborhood.

Here is a link to the notes:

http://math.stanford.edu/~vakil/216blog/FOAGjun2711publicimperfect.pdf

Take a look at chapter 14.

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+1 for the link, Lalit. I keep forgetting that there are users who don't know these notes yet: they are in for a very, very nice surprise! –  Georges Elencwajg Jul 21 '11 at 18:18

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