Analytic variety can be defined by finitely many equations Let $(f_i)_{i\in I}$ be a family of holomorphic functions on $\mathbb C^n$. $V:=\{z\in\mathbb C^n\mid f_i(z)=0,\forall i\}$. I know that locally we could write down the equation with finitely many equations because the local ring $\mathscr{O}_0$ of the analytic sheaf is Noetherian.
But it this also true globally? First, we know that the ring of all holomorphic functions on $\mathbb C^n$ is not Noetherian.
I attempted like this assume $I_0(V):=\{f\in \mathscr{O}_0\mid f|_V=0\}$ is an ideal of  $\mathscr{O}_0$, so it is finitely generated say by $(g_1,...,g_m)$. Then when we localize any $f_i$ to $\mathscr{O}_0$, we can write $f_{i,x}=r_1g_1+...+r_mg_m$. This means they coninside on  an open neighborhood. Then apply identity theorem. But later when I reviewed it, it was not that correct, because the functions $g_1,r_1,...,r_m,g_m$ might actually not be globally defined. So maybe this claim is wrong? is there any counter example?

Yes, this is true, even if you replace ${\mathbb C}^n$ by an $n$-dimensional complex manifold $M$. The precise statement is that if $V\subset M$ is the zero-set of a family of holomorphic functions $f_j, j\in J$, on $M$ then there are  holomorphic functions $g_1,...g_n$ on $M$ such that
$$
V=\{z\in M: g_i(z)=0, i=1,...,n\}. 
$$
See Proposition 5.7 in
E.M. Chirka, "Complex analytic sets." Translated from the Russian by R. A. M. Hoksbergen, Mathematics and Its Applications: Soviet Series, 46. Dordrecht etc.: Kluwer Academic Publishers. xix, 372 p. Dfl. 195.00; (1989).
