On a splitting problem (Q1865016)

Let \(\Omega\) be a complex manifold modelled on a separable Fréchet space \(X\). In this paper, the author proves that if the sheaf cohomology groups \(H^q (\Omega , \mathcal{O})\) vanish for \(q=1,\ldots,n-m\) and the \(m \times n\) matrix \(f(x)=(f_{ik}(x))\) has constant rank \(m\) on \(\Omega\), then there are \(g_{kj}\in \mathcal{O}(\Omega)\) such that on \(\Omega\), \[ \sum_{k=1}^n f_{ik}g_{kj}=\delta_{ij}. \] By a theorem of \textit{L. Lempert} [Invent. Math. 142, 579-603 (2000; Zbl 0983.32010)] the cohomology groups \(H^q (\Omega , \mathcal{O})\) vanish for all \(q\geq 1\) if \(\Omega\) is a pseudoconvex open subset of a Banach space with unconditional basis. This, together with a particular case of the main theorem (\(m=1\)), answers in the affirmative and for a wide range of spaces \(X\), the following question: If \(\Omega \subset X\) is a pseudoconvex open set, does every proper finitely generated ideal of \(\mathcal{O}(\Omega)\) have a common zero? This generalizes a recent result of \textit{J. Mujica} [Arch. Math. 76, 292-298 (2001; Zbl 0993.46026)].