[proofplan] We prove the domain-of-holomorphy form of Cartan's Theorem B by combining the Levi problem, the Dolbeault resolution, and a finite free resolution argument for coherent analytic sheaves. The analytic input is the vanishing of Dolbeault cohomology on pseudoconvex domains. Coherence then lets us reduce arbitrary coherent sheaves to free sheaves locally and pass through the long exact cohomology sequence. [/proofplan] [step:Reduce the domain of holomorphy to the pseudoconvex analytic setting] Since $\Omega$ is a domain of holomorphy, the solution of the Levi problem implies that $\Omega$ is pseudoconvex. On pseudoconvex domains in $\mathbb{C}^n$, Hörmander's $L^2$ estimates for $\bar\partial$ give solvability of the equation $\bar\partial u=f$ for every $\bar\partial$-closed $(0,q)$-form with $q\ge 1$, after the usual exhaustion and approximation argument. Therefore \begin{align*} H^{0,q}_{\bar\partial}(\Omega)=0 \qquad (q\ge 1). \end{align*} By the Dolbeault theorem, this is equivalent to $H^q(\Omega,\mathcal{O}_\Omega)=0$ for $q\ge 1$. [/step] [step:Extend the vanishing from $\mathcal{O}$ to finite free sheaves] For every finite free sheaf $\mathcal{O}_\Omega^r$, cohomology commutes with finite direct sums, so \begin{align*} H^q(\Omega,\mathcal{O}_\Omega^r) \cong H^q(\Omega,\mathcal{O}_\Omega)^r=0 \end{align*} for every $q\ge 1$. Thus all finite rank locally free analytic sheaves that are globally trivial on the domain have vanishing higher cohomology. This is the base case for the coherent sheaf argument. [/step] [step:Use coherence to resolve $\mathcal{F}$ by finite free sheaves] The sheaf $\mathcal{F}$ is coherent. Hence locally it admits finite presentations by free $\mathcal{O}_\Omega$-modules, and on a Stein domain these local presentations can be assembled into a finite locally free resolution on the sets of a Stein exhaustion. Equivalently, one may use Cartan's syzygy theorem: there is an exact complex \begin{align*} 0 \to \mathcal{K}_m \to \mathcal{O}^{r_{m-1}} \to \cdots \to \mathcal{O}^{r_0} \to \mathcal{F} \to 0 \end{align*} with coherent kernels. Applying the long exact sequence in sheaf cohomology to the short exact sequences obtained from this resolution reduces vanishing for $\mathcal{F}$ successively to vanishing for finite free sheaves. [/step] [step:Conclude by dimension shifting] Let $0\to \mathcal{K}\to \mathcal{O}^r\to \mathcal{F}\to 0$ be the first finite presentation. Since $H^q(\Omega,\mathcal{O}^r)=0$ for $q\ge 1$, the long exact sequence gives isomorphisms \begin{align*} H^q(\Omega,\mathcal{F}) \cong H^{q+1}(\Omega,\mathcal{K}) \qquad (q\ge 1). \end{align*} The kernel $\mathcal{K}$ is coherent, so the same argument applies to $\mathcal{K}$. Iterating through a finite coherent resolution gives $H^q(\Omega,\mathcal{F})=0$ for every $q\ge 1$. This is Cartan's Theorem B. [/step]