Abelianizing the divided powers algebra of an algebra
type: Item
instance of: scholarly article
title: Abelianizing the divided powers algebra of an algebra
zbMATH Open document ID: 0692.16004
DOI: 10.1016/0021-8693(89)90215-9
author: Dieter Ziplies
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let R be a commutative unital ring and let A and B be R-modules. Then the polynomial laws from A to B may be represented as special R-module homomorphisms from \(\Gamma\) (A) to B where \(\Gamma\) (A) is the divided powers algebra of A [\textit{N. Roby}, Ann. Sci. Ec. Norm. Supér., III. Sér. 80, 213-348 (1963; Zbl 0117.023), IV.1 and IV.2]. If, in addition, A and B carry an R-algebra structure as well, those polynomial laws from A to B which respect the multiplicative structure may be represented by R-algebra homomorphisms from a certain R-algebra \({\hat \Gamma}\)(A) to B [the author, ``Dividierte Potenzen, Determinanten und die Algebra der verallgemeinerten Spurpolynome'' (Diss. Univ. Düsseldorf, 1983), 1.7.6]. The structure of \({\hat \Gamma}\)(A) is usually very complex but if B is commutative then one may restrict to \(C={\hat \Gamma}(A)/[{\hat \Gamma}(A)]\) which is a commutative algebra. For example if A is the free algebra then C is the algebra of commutative trace monomials [the author, Beitr. Algebra Geom. 24, 9-27 (1987; Zbl 0632.16004)]. The main result of the paper is the fact that if \(A=(\oplus_ iB_ i)\oplus [A,A]\) (sum of R-modules) where the \(B_ i\) are commutative R-subalgebras of A and [A,A] is the linear span of the commutators then \(C=\oplus_ i{\hat \Gamma}(B_ i)\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 4132374
zbMATH Keywords: polynomial laws
zbMATH Keywords: R-module homomorphisms
zbMATH Keywords: divided powers algebra
zbMATH Keywords: algebra of commutative trace monomials
zbMATH Keywords: commutators
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90215-9
OpenAlex ID: W2041510993
cites work: Formen auf Moduln über kommutativen Ringen beliebiger Charakteristik.
cites work: The invariant theory of \(n\times n\) matrices
cites work: Lois polynomes et lois formelles en théorie des modules
cites work: Q3322254
cites work: Divided powers and multiplicative polynomial laws
cites work: A characterization of the norm of an Azumaya algebra of constant rank through the divided powers algebra of an algebra
cites work: Generators for the divided powers algebra of an algebra and trace identities
description: scientific article; zbMATH DE number 4132374