Let $\Omega$ be a Stein manifold and $V\subset \Omega$ an analytic set. Show that para every $z\in \Omega\setminus V$ there is a holomorphic function $f$ on $\Omega$ vanishing on $V$ and $f(z)\neq0$.
If $\Omega$ is Stein, it is holomorphically convex, I think that by taking some point out of a compact set I can get a function.
Could you give me any hints?
Thank you!
Since $\Omega$ is Stein, by Cartan's Theorem A, ideal sheaf $I_V$ is globally generated, namely there is a surjection
$$\mathcal{O}_{\Omega}^{I}\twoheadrightarrow I_{\Omega}.$$
So $\Omega$ is zero loci of a family of holomorphic functions $\{f_i\}_{i\in I}$ defined on $\Omega$.
In particular, for any $x\in \Omega\setminus V$, there is an $i\in I$ such that $f_i(x)\neq 0$.