Scalar holomorphic sheaf acyclicity on pseudoconvex domains

On each coordinate polydisc in $\mathrm{\Omega }$, [F2] says the sequence of sheaves $0\to {\mathcal{O}}_{\mathrm{\Omega }}\to {\mathcal{A}}^{0,0}\stackrel{\bar{\mathrm{\partial }}}{\to }{\mathcal{A}}^{0,1}\stackrel{\bar{\mathrm{\partial }}}{\to }\cdots$ is exact in positive degree. Its degree-zero kernel is exactly ${\mathcal{O}}_{\mathrm{\Omega }}$: a smooth function in that kernel has all $\bar{z}$-derivatives zero and is holomorphic by [F3]. Thus it is an exact sheaf resolution on $\mathrm{\Omega }$.

Each resolution term is a sheaf of modules over smooth functions and is $\mathrm{\Gamma }(\mathrm{\Omega },-)$-acyclic by [F4]. The acyclic-resolution theorem [F5] therefore identifies $H^q(\mathrm{\Omega },{\mathcal{O}}_{\mathrm{\Omega }})$ with the cohomology in degree $q$ of the global smooth Dolbeault complex.

For $1\le q\le m$, [F6] makes every closed global smooth $(0,q)$-form exact, so the degree-$q$ cohomology in step 2.1 is zero. For $q>m$, the complex has no $(0,q)$-forms on an $m$-dimensional complex manifold, so its degree-$q$ cohomology is zero as well. Hence $H^q(\mathrm{\Omega },{\mathcal{O}}_{\mathrm{\Omega }})=0$ for every $q>0$.