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Thread: Hartshorne Question

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    Hartshorne Question

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

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

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

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

    where the left $\displaystyle \text{Hom}$ means morphisms of varieties, and the right $\displaystyle \text{Hom}$ means homomorphisms of $\displaystyle k$-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; Mar 25th 2010 at 04:56 PM.
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