Smooth algebras
type: Item
instance of: scholarly article
title: Smooth algebras
zbMATH Open document ID: 0604.16015
DOI: 10.1016/0021-8693(86)90160-2
author: William F. Schelter
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The problem underlying this paper is the search for an appropriate analogue, in the setting of prime P.I. algebras over an algebraically closed field k, of the notion of a nonsingular variety. Such an algebra A is here called smooth if, whenever A occurs as the factor of a k-algebra C by an ideal N of square zero, there is a copy of A in C splitting the extension of N by A. Moreover, C should only be allowed to come from a category of algebras satisfying the same identities as A. The main result proved here uses [\textit{A. Brumer}, Bull. Am. Math. Soc. 69, 721-724 (1963); addendum 70, 185 (1964; Zbl 0113.260)] to demonstrate that this is a sensible definition in dimension 1, by showing that if A is an algebra as above, and A has Krull dimension 1, then A is smooth (in the category of associative k-algebras) if and only if it is smooth in the category of k-algebras satisfying the same identities as A if and only if all right ideals of A are projective if and only if it is geometrically integrally closed. The last concept here is defined in [\textit{M. Artin} and \textit{W. Schelter}, Adv. Math. 39, 289-329 (1981; Zbl 0461.16014)].
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 3975143
zbMATH Keywords: finite global dimension
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: prime P.I. algebras
zbMATH Keywords: category of algebras
zbMATH Keywords: identities
zbMATH Keywords: Krull dimension
zbMATH Keywords: geometrically integrally closed
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90160-2
OpenAlex ID: W4212848934
cites work: Integral ring homomorphisms
cites work: On the Global Dimension of Some Filtered Algebras
cites work: On the cohomology groups of an associative algebra
cites work: Q5343374
cites work: Q4061092
cites work: Integral extensions of rings satisfying a polynomial identity
description: scientific article; zbMATH DE number 3975143