(i) $\mathcal{O}_X$ is globally (hence locally) finitely generated by the single global section $1 \in \Gamma(X, \mathcal{O}_X)$.
(ii) The surjective sheaf morphism $\mathcal{O}_X \to \mathcal{O}_X$ sending $1 \mapsto 1$ is globally an isomorphism, so a fortiori the kernel is locally finitely ge...
I don't understand what you are trying to prove here: are you trying to show that $\mathscr O_X$ is always a coherent $\mathscr O_X$-module? I think this is not true in general.
I cannot understand the proof: Say I have a morphism $u:\mathscr O_X^n|_U\rightarrow \mathscr O_X|_U;$ how did you propose to prove that $\operatorname{Ker}(u)$ is locally finitely generated?
I am currently trying to read EGA; and I was confronted with a proposition that, hypotheses being as in the question, an $\mathscr O_X$-module is coherent if and only if it is $\cong \tilde M$ where $M$ is a finitely generated $A$-module. I cannot uderstand the last step of that proof, but I can complete it if I can show the statement in question. Also, it is mentioned that $\mathscr O_X$ is not necessarily a coherent $\mathscr O_X$-module.
Well, if you have a morphism $u:\mathscr O_X^n|_U\rightarrow \mathscr O_X|_U$, then you have a short exact sequence $0 \to \ker u \to \mathscr O_X^n|_U \xrightarrow{u} \mathscr O_X|_U \to 0$, but since $\mathscr O_X|_U$ is a projective $\mathscr O_X|_U$-module, the sequence splits, so $\ker u$ is a summand of $\mathscr O_X^n|_U$, hence is a finitely generated $\mathscr O_X|_U$-module
I think here the morphism u is a morphism of sheaves; and I am not familiar with the notion of projective sheaf of modules. Maybe you can explain more? Thanks.
Ah, I was under the impression that it was a morphism of modules over the ring O_X(U). For me the proposition that you're trying to prove is the definition of coherent
However, O_X|_U-modules form an abelian category, in which O_X|_U is a free object, so the sequence should still split
Very well; I won't argue with your definition. Do you understand the argument though? The idea is to show that any sequence of O_X|_U-modules, where the last term is O_X|_U, splits
For example, in page 17 of this [pdf file](http://stacks.math.columbia.edu/download/modules.pdf), they assume that $O_X$ is a coherent $O_X$ module; this would not be needed if $O_X$ is always a coherent $O_X$-module. Of ocurse this is an indirect evidence.
I don't think your link is evidence though: in that statement they don't assume X is a scheme, only a ringed space. My claim is only for schemes (if that wasn't clear)
http://math.stackexchange.com/questions/52856/is-noetherian-condition-always-needed-when-speaking-of-a-coherent-sheaf Check the first answer! "The problem is that coherence is very difficult to check in general and actually for some schemes, even affine ones, the structure sheaf OX is not coherent, and in that sad case the concept coherent is essentially worthless ."
Sorry for the impolite tone; I was too excited to find this statement.
No worries - no offense taken. Thank you for finding, and linking to, Georges' excellent answer
It seems my mistake was that I thought the morphism u had to be surjective. That is apparently not the case, and consequently I retract all previous statements I made
So, if you believe the equivalence between coherent sheaves and finitely generated quasi-coherent sheaves on a locally Noetherian scheme (as in Georges' answer) then you have an answer to your problem, although it may not be as direct as you were hoping for
Discussion between awllower and zcn
Discussion between awllower and zcn