State spaces, finite algebras, and skew group rings
type: Item
instance of: scholarly article
title: State spaces, finite algebras, and skew group rings
zbMATH Open document ID: 0727.16004
DOI: 10.1016/0021-8693(91)90286-H
author: Ricardo Alfaro
published in: Journal of Algebra
publication date: 1991-01-01T00:00:00Z
review text: The state space, St(R), of a ring R is the set of all additive maps s: \(K_ 0(R)\to {\mathbb{R}}\) which send [R] to 1 and \(K^+_ 0(R)\) to \({\mathbb{R}}_{\geq 0}\). St(R) is a compact convex subset of \({\mathbb{R}}^{K_ 0(R)}\), equipped with the product topology, and hence, by the Krein- Milman Theorem, St(R) is determined by the set \(\partial_ e St(R)\) of its extreme points. After establishing some general properties of St(R) for a Noetherian ring R, such as the behavior under localization at Ore subsets of R, the present paper focusses on the case where \(R=S*G\) is the skew group ring of the finite group G acting faithfully on the commutative Noetherian domain S with \(| G|^{-1}\in S\). For example, for S regular of finite Krull dimension, it is shown that \(\partial_ e St(S*G)\) is contained in the set of so-called generalized rank functions associated with maximal ideals of S*G. For S a Dedekind domain, the complete topological structure of St(S*G) is determined. This latter part depends on work of \textit{K. R. Goodearl} and \textit{R. B. Warfield} [J. Algebra 71, 322-378 (1981; Zbl 0472.16005)].
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 4200405
zbMATH Keywords: state space
zbMATH Keywords: compact convex subset
zbMATH Keywords: Krein-Milman Theorem
zbMATH Keywords: extreme points
zbMATH Keywords: Noetherian ring
zbMATH Keywords: localization
zbMATH Keywords: Ore subsets
zbMATH Keywords: skew group ring
zbMATH Keywords: Krull dimension
zbMATH Keywords: generalized rank functions
zbMATH Keywords: Dedekind domain
reviewed by: Martin Lorenz
MaRDI profile type: Publication
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full work available at URL: 0021-8693(91)90286-h
OpenAlex ID: W2015795306
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description: scientific article; zbMATH DE number 4200405