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Math Help - Hartshorne, corollary question

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    Hartshorne, corollary question

    If X, Y are two affine varieties, then X and Y are isomorphic if and only if A(X) and A(Y) are isomorphic as k-algebras.

    In my textbook (Hartshorne), it says that this immediately follows from a proposition that states:


    Let X be any variety and let Y be an affine variety. Then there is a natural bijective mapping of sets

    \alpha : \text{Hom}(X, Y) \overset{\sim}{\rightarrow} \text{Hom}(A(Y), \mathcal{O}(X))

    where the left \text{Hom} means morphisms of varieties, and the right \text{Hom} means homomorphisms of k-algebras.

    How is the proof of this immediate from the proposition? I do not see why. Thanks.
    Last edited by pascal4542; March 25th 2010 at 04:54 PM.
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