On certain homological properties of Stein algebras
type: Item
instance of: scholarly article
title: On certain homological properties of Stein algebras
zbMATH Open document ID: 0944.46048
DOI: 10.1007/BF02169288
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-01-25T00:00:00Z
review text: A Fréchet algebra is called a Stein algebra if it is topologically isomorphic to an algebra of holomorphic functions \({\mathcal O}(X)\) on some complex Stein space \((X,{\mathcal O}_X)\). In this paper two problems are considered concerning homological properties of Stein algebras: 1) Is it true that any projective Fréchet module over an algebra of holomorphic functions \({\mathcal O}(U)\) on a domain \(U\subset \mathbb{C}^n\) is free (i.e. topologically isomorphic to a module \({\mathcal O}(U,E)\) of all holomorphic functions on \(U\) with values in the Fréchet space \(E\))? 2) Does the global homological dimension of a Fréchet algebra of holomorphic functions on a complex Stein manifold \(X\) coincide with the dimension of \(X\)? The author shows 1) that all finitely generated projective Fréchet \({\mathcal O}(U)\)-modules are free if and only if all vector bundles over \(U\subset \mathbb{C}^n\) are trivial, 2) the weak (= flat) homological dimension of a Fréchet algebra of holomorphic functions on a complex Stein manifold coincides with the dimension of the manifold.
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46M18
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 1393016
zbMATH Keywords: Fréchet algebra
zbMATH Keywords: Stein algebra
zbMATH Keywords: algebra of holomorphic functions
zbMATH Keywords: complex Stein space
zbMATH Keywords: projective Fréchet module
zbMATH Keywords: global homological dimension
zbMATH Keywords: vector bundles
author: A. Yu. Pirkovskii
MaRDI profile type: Publication
cites work: Big projective modules are free
cites work: Q3843687
cites work: Zur Theorie der Steinschen Algebren und Moduln
cites work: Analytische Faserungen über holomorph-vollständigen Räumen
cites work: Q3871181
cites work: Q5515607
cites work: Coincidence of the homological dimensions of the Fréchet algebra of smooth functions on a manifold with the dimension of the manifold
cites work: Q3897512
cites work: Projective modules over polynomial rings
cites work: Faisceaux algébriques cohérents
cites work: Q4124942
cites work: Homology and cohomology for topological algebras
cites work: A general framwork for a multi-operator functional calculus
full work available at URL: https://doi.org/10.1007/bf02169288
OpenAlex ID: W2169814103
Recommended article: Q5313762
Recommended article: Q4761618
Recommended article: Uniform Fréchet algebras with prescribed endomorphism group
Recommended article: Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds
Recommended article: Finite Codimensional Subalgebras of Stein Algebras and Semiglobally Stein Algebras
Recommended article: Q3223127
Recommended article: Q3743597
Recommended article: Global construction of the normalization of Stein spaces
Recommended article: Towards Oka-Cartan theory for algebras of holomorphic functions on coverings of Stein manifolds. I.
Recommended article: EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE
description: scientific article; zbMATH DE number 1393016