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Suppose that $V$ is a variety over a field $k=\overline{k}$,and $P\in V$,$\mathcal{O}_P$ is the local ring of $P$ on $V$.Is it in general that $\mathcal{O}_P$ is not a finitely generated $k$ algebra?I know that $\mathcal{O}_P\cong A(Y)/\mathfrak{m}_P$,yet I am still not clear how to get the result. Will someone be kind enough to help me figure this out in detail?Thank you very much!

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    Have you tried any examples?2011-08-13
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    @Qianchu Yuan:I tried the simplest case:take V to be k^n.Yet in this case Op is k,f.g.over k.2011-08-13
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    You aren't using the definition of "local ring of $P$" that I think is standard (what you describe is the _residue field_). You should probably review the definition in whatever text you're working with.2011-08-13
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    @Qiaochu Yuan:Err,you are right.I got the definition wrong.Thank you very much!2011-08-13

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