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Motivation: I am working on problem II.5.15 in Hartshorne's Algebraic Geometry, which is to prove that, given a noetherian scheme $X$, an open subset $U\subset X$, and a coherent sheaf $\mathscr{F}$ on $U$, there exists a coherent sheaf on $X$ restricting to $\mathscr{F}$ on $U$. To shape my thinking, I would like to be working with some examples; I would like help in coming up with examples that reveal the situation's subtleties.

My question: Let $X=\operatorname{Spec}A$ be a noetherian affine scheme.

Can you give an explicit example of an open set $U\subset X$ that is not affine, and a coherent sheaf $\mathscr{F}$ on $U$ that does not in an obvious way come from a module over $A$?

Apologies for the imprecision in the term "in an obvious way." For the sake of clarification, here's an example of what I'd like: let $A=k[x,y]$ and $U=\operatorname{Spec}A \setminus (0,0) = D(x)\cup D(y)$. I'd like a finitely generated module $M$ over $\mathcal{O}_X(D(x))=k[x,x^{-1},y]$ and another one $N$ finitely generated over $\mathcal{O}_X(D(y))=k[x,y,y^{-1}]$ such that $M_y\cong N_x$, but not because $M$ and $N$ are both visibly localizations of a module over $k[x,y]$ in the first place. (I understand that because pushforwards of quasicoherent sheaves on noetherian schemes are quasicoherent, $M$ and $N$ will, in fact, necessarily be localizations of some module over $A$; I'm just going for how they're presented to me not making this obvious, so that I have to do the work of figuring out how they fit together.)

Thanks in advance!

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I think you want a coherent sheaf $F$ on a quasi-affine scheme $U$ such that $F\ne M^{\sim}$ for any $O(U)$-module $M$. –  user18119 Apr 8 '13 at 9:18
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