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Let $X$ be a Stein complex analytic space, and let $Z$ be a closed complex analytic subspace. Set $U=X-Z$.

I was wandering if there is any relationship between $A_1:=\mathcal{O}_X(U)$

and the localization (in the purely algebraic sense) $A_2:=\mathcal{O}_X(X)_f$, where $f$ is a global holomorphic function such that $f \cdot \mathcal{O}_X$ is the ideal sheaf of $Z$ (if I'm not mistaken, such an $f$ exists, anyway assume we're in the case in which it exists for the purpose of this question).

Is $A_1$ perhaps some sort of "completion" of $A_2$ ?

What happens in the toy example $X=\mathbb{C}$, $Z=$ {$0$}, $f(z)=z$?

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  • $\begingroup$ See Kisin's paper "Analytic functions on Zariski open sets...", especially Prop. 1.3.1, for the appropriate kind of completion to use. (The existence of $f$ as in the question must be assumed.) $\endgroup$
    – BCnrd
    Commented Apr 23, 2010 at 15:14
  • $\begingroup$ Thanks. Any online not subscription-requiring versions of that paper? $\endgroup$
    – Qfwfq
    Commented Apr 23, 2010 at 15:51
  • $\begingroup$ @unknown: go to Kisin's webpage. $\endgroup$
    – BCnrd
    Commented Apr 23, 2010 at 16:21

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