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I'm studying affine algebraic varieties(or also called closed sets). I'm using the book of Shafarevich. Let's assume that $k=\overline{k}$

There is a proposition that said:

Let $X,Y$ be two closed sets in the affine space $k^n$ , then:

$X,Y$ are isomorphic (i.e there exist a regular map , with a inverse regular map) if and only if:

$K[X],K[Y]$ (they coordinate rings) are isomorphic as k-algebras.

I only proved the first side ( $X,Y$ isomorphic $\Rightarrow$ $K[X],K[Y]$ isomorphic

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Do you know that there is a natural bijection between regular maps $X \to Y$ and algebra homomorphisms $k[Y] \to k[X]$? –  Zhen Lin Sep 16 '12 at 4:03
@Daniel : What do you want to ask ? Please make it precisely. I guess that you want to prove the reverse : If the coordinate rings $K[X], K[Y]$ are isomorphic, then $X$ and $Y$ are isomorphic ? –  Nguyễn Duy Khánh Sep 19 '12 at 3:04
@NguyễnDuyKhánh Yes man! exactly that :D! –  Daniel Sep 24 '12 at 17:31

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