Localization, catenarity and dimensions in rings equipped with a Lie algebra action
The objects of study in this paper are smash products \(R \# U ({\mathfrak g})\), where \(R\) is a Noetherian algebra, (often commutative), over a field \(k\) and \({\mathfrak g}\) is a (usually solvable or nilpotent) Lie algebra of finite dimension over \(k\) acting as derivations on \(R\). The topics treated include heights of prime ideals, catenarity, localisation of prime ideals, and homological conditions such as the finiteness of the global or injective dimensions of \(R \# U ({\mathfrak g})\). It is often necessary to assume that \({\mathfrak g}\) acts hypernormally or hypercentrally on \(R\). The results obtained generalise some well-known results for \(U ({\mathfrak g})\), and also the case of commuting derivations treated in [\textit{K. A. Brown}, \textit{K. R. Goodearl} and \textit{T. H. Lenagan}, Trans. Am. Math. Soc. 317, 749-772 (1990; Zbl 0692.16002)]. (Also submitted to M. R.).
scientific article; zbMATH DE number 826740