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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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    @NguyễnDuyKhánh Yes man! exactly that :D!2012-09-24

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