Abelianizing the divided powers algebra of an algebra
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 \(C=\oplus_ i{\hat \Gamma}(B_ i)\).
scientific article; zbMATH DE number 4132374