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Suppose $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ are isomorphic as schemes. Then by definition there is an isomorphism of locally ringed spaces $(\psi, \psi^{\sharp}): (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$.

Now, I would like to be able to use $(\psi, \psi^{\sharp})$ to identify $X$ with $Y$ -- that is, convert all questions about the scheme-theoretic structure -- the topology and structure sheaf -- of $X$ to questions about the corresponding structure of $Y$. But here's the problem: as far as I know, there's no guarantee that the isomorphism of sheaves $\psi^{\sharp}$ will induce an isomorphism on the ring of sections over an open set! (Recall that surjectivity may fail on sections.) So, for all that I can tell, we might very well have a situation where $X$ is isomorphic to $\rm{Spec} \, \, A$ for some ring $A$, and yet $\Gamma(X, \mathcal{O}_X) \not \cong A$!

Something tells me this can't be right. What, if anything, am I missing?

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    Charles, thanks; that's exactly what I was missing.2011-04-22

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