TITLE: Is Noetherian condition always needed when speaking of a coherent sheaf?
QUESTION [10 upvotes]: 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.

REPLY [7 votes]: 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.