Reference for a theorem related to second Cousin problem

Question

I need a reference for the following theorem:

Let $\Omega\subseteq \mathbb{C}^n$ be a domain of holomorphy. Let $\mathcal{O}^*$ be the sheaf of germs of invertible holomorphic functions on $\Omega$. Then $$\check{\mathrm{H}}^1(\Omega,\mathcal{O}^*)=0\iff\check{\mathrm{H}}^2(\Omega,\mathbb{Z})=0.$$

Thank you.

Answer 1

The exponential sequence for $\Omega$ gives an exact sequence \begin{align*} \cdots \check H^1(\Omega, {\cal O}) \to \check H^1(\Omega, {\cal O}^*) \to \check H^2(\Omega, \mathbb{Z}) \to \check H^2(\Omega, {\cal O}) \to \cdots \end{align*} But ${\cal O}$ is coherent over $\Omega$ and thus acyclic by Cartan's Theorem B.