# Complex spaces with a global holomorphic function being a zero divisor in a local ring

Is there a simple example of a complex space $X$ with the following property: there is a function $f$ holomorphic on $X$ such that: (i) $f$ is not a zero divisor in the ring ${\mathcal O}(X)$ of all functions holomorphic on $X$; (ii) $f$ is a zero divisor in the local ring ${\mathcal O}_x(X)$ for some point $x\in X$?

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What's your motivation for asking this, if I may be so bold? –  David Roberts Jun 25 '12 at 6:02
That would take a bit too long to explain in detail, but, really, it has something to do with how one should define meromorphic functions on an arbitrary complex space. –  Alex Jun 25 '12 at 6:15
I guess the first example in S. Kleiman, Misconceptions about $K_X$ (L'Enseignement Mathématique, 25(1979), 203-206) still works. –  user2035 Jun 25 '12 at 7:15
I have known about that example for a while, and my question was in fact motivated by it. In that example one starts with the projective line over a domain with nonzero maximal ideal. But how can you produce an example of a complex space from it? I have to work over a field. –  Alex Jun 25 '12 at 21:57
As suggested in the paper, $A$ can be a finitely generated algebra over $\mathbf C$. –  user2035 Jun 27 '12 at 12:14