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Given a complex algebraic, affine variety $X$ and a subvariety $V$, let $I_V$ indicate the ideal of polynomials on $X$ with vanish on $V$.

Given an open subset $U$ of $X$, is it true that the ideal of holomorphic functions on $U$ which vanish on $V\cap U$ is generated by $I_V$?

flag	asked Jan 24 at 1:24 Brett Parker 140●6

	holomorphic functions will probably not be generated by polynomials. did you mean regular functions on $U$? – Sándor Kovács Jan 24 at 3:13
	To clarify, I meant that every holomorphic function on U which vanishes on $V$ may be written as a sum of holomorphic functions times polynomials in $I_V$. – Brett Parker Jan 24 at 3:30
	I see. That makes more sense. Perhaps you should edit the question...(say add the ring in which you are working). – Sándor Kovács Jan 24 at 8:25

Brett Parker · Answer 1 · 2013-01-24 04:03:06Z UTC

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This is true. Alex Isaev pointed out to me that it is Theorem 4.6 from `A geometric criterion for algebraic varieties' by Rudin. A local version is proposition 4 of Serre's GAGA paper.

edited Jan 24 at 4:03

answered Jan 24 at 3:27

Brett Parker
140●6

Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

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Can a locally defined holomophic function which vanishes on a subvariety $V$ be written in terms of globally defined polynomials vanishing on $V$?

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