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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 $I_V$?

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holomorphic functions will probably not be generated by polynomials. did you mean regular functions on $U$? –  Sándor Kovács Jan 24 '13 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 '13 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 '13 at 8:25
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1 Answer

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.

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