Easy question about coordinate rings

Asked Feb 12 '14 at 14:11

Active Feb 12 '14 at 14:11

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Let $C=V(x^2+y^2-1)$ be an affine algebraic curve. In an online course the professor said $\varphi=\frac{x-1}{y}\in A(C)$, but he didn't explain why.

I would like to know how he gets this function and why it's in the coordinate ring of the curve $C$.

Thanks in advance

asked Feb 12 '14 at 14:11

user42912

It's not true. The function blows up at $(-1, 0)$. – Zhen Lin Feb 12 '14 at 14:53
@ZhenLin It's regular at the point $(1,0)$, I think the professor meant at this point. – user42912 Feb 12 '14 at 15:38
Functions in the coordinate ring are regular everywhere, however. – Zhen Lin Feb 12 '14 at 15:42

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