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

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 $I_V$?

-
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