On certain homological properties of Stein algebras
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.
scientific article; zbMATH DE number 1393016
scientific article