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How to find the coordinate ring of $Z(x^{2}+y^{2}-1)$? (Here $k$ is an algebraically closed field with $\operatorname{char}k\neq 2$.)

I'm not sure how to proceed. Can you please help? I'm trying to find out if we can prove/disprove if the coordinate ring is isomorphic to $k[x,x^{-1}]$.

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up vote 5 down vote accepted

For the second part of your question, it is true that $k[x,y]/(x^2+y^2-1)$ is isomorphic to $k[t,t^{-1}]$. For instance, let $i$ be one of the square roots of $-1$ in the algebraically closed field $k$, and send $t$ to $y-ix$, $t^{-1}$ to $y+ix$. The inverse isomorphism sends $x$ to $i(t-t^{-1})/2$ and $y$ to $(t+t^{-1})/2$.

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By definition, the coordinate ring is $k[x,y]/(x^2+y^2-1)$.

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Maybe we should be a little more careful. We have some algebraic set $Y$ and we want to find $k[x, y]/I(Y)$, where $I(Y)$ is the ideal of $Y$. Certainly $(x^2 + y^2 - 1) \subset I(Y)$, but equality here doesn't seem to be a matter of definition. [And maybe the OP should specify what $k$ is.] – Dylan Moreland Feb 22 '12 at 16:09
I am sorry. You are right. – Yang Zhou Feb 24 '12 at 1:58

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