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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? – Qiaochu Yuan Aug 13 '11 at 14:54
@Qianchu Yuan:I tried the simplest case:take V to be k^n.Yet in this case Op is k,f.g.over k. – user14242 Aug 13 '11 at 15:08
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. – Qiaochu Yuan Aug 13 '11 at 15:11
@Qiaochu Yuan:Err,you are right.I got the definition wrong.Thank you very much! – user14242 Aug 13 '11 at 15:32

The residue field is finitely generated as soon $A(Y)$ is: this is immediate since the former is a quotient of the latter.

On the other hand, the local ring at a point is very rarely finitely generated as an algebra. Consider the very simple example in which $Y$ is the affine line and the point is the origin: if you are able to describe the local ring explicitly, then you can easily show it is not f.g.

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