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Math Help - Hartshorne Question

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    Hartshorne Question

    The functor X \rightarrow A(X) induces an arrow-reversing equivalence of categories between the category of affine varieties over k and the category of finitely generated integral domains over k.

    My textbook (Hartshorne) says that this is a corollary to the following proposition:

    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.

    I do not see how to apply the proposition to prove this corollary. I would appreciate advice on this. Thanks.
    Last edited by pascal4542; March 25th 2010 at 05:56 PM.
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