On a splitting problem
type: Item
instance of: scholarly article
title: On a splitting problem
zbMATH Open document ID: 1017.46029
DOI: 10.1016/S0007-4497(02)01129-6
author: Imre Patyi
published in: Bulletin des Sciences Mathématiques
publication date: 2003-03-23T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 16E45
zbMATH DE Number: 1886872
zbMATH Keywords: ideals of holomorphic functions
zbMATH Keywords: splitting problem
reviewed by: Ignacio Zalduendo
MaRDI profile type: Publication
cites work: Q3692032
cites work: The Dolbeault complex in infinite dimensions. III: Sheaf cohomology in Banach spaces
cites work: Analytic cohomology in Fréchet spaces
cites work: Ideals of holomorphic functions on Tsirelson's space
cites work: Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques
description: scientific article; zbMATH DE number 1886872