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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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You are missing having spent the time to try to show that when you have an isomorphim of schemes that does not happen. Try to do it: if you do get stuck, them you can ask for help in unstucking yourself! –  Mariano Suárez-Alvarez Apr 22 '11 at 19:57
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If your definition of isomorphism of schemes doesn't imply this, then you're using the wrong definition of an isomorphism of schemes. –  Qiaochu Yuan Apr 22 '11 at 19:57
    
Qiaochu, yes, that's what I meant by "something tells me that can't be right"! –  kiuscias Apr 22 '11 at 20:31
    
Mariano, the sticking point is the fact that while a surjection of sheaves induces a surjection on stalks, it doesn't necessarily induce a surjection on sections. There must be information contained in the definition of a morphism of schemes that somehow says $\psi^{\sharp}$ is more than just an arbitrary morphism of sheaves, but the only other condition explicitly involved is that it has to a induce local morphisms on the stalks, and I don't see what that could have to do with it. –  kiuscias Apr 22 '11 at 20:40
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If you know that a morphism of sheaves is injective, and it induces surjections on stalks, you can use injectivity to show that it induces surjections on sections. –  Charles Staats Apr 22 '11 at 21:53
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