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!
Is $\mathcal{O}_P$ a finitely generated algebra over k?
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algebraic-geometry
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0@Qiaochu Yuan:Err,you are right.I got the definition wrong.Thank you very much! – 2011-08-13
1 Answers
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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.