Derived functors of the divided power functors

Abstract

We study the derived functors of the components ${\Gamma }_{\mathbb{Z}}^d\left(A\right)$ of the divided power algebra ${\Gamma }_{\mathbb{Z}}\left(A\right)$ associated to an abelian group $A$, with special emphasis on the $d=4$ case. While our results have applications both to representation theory and to algebraic topology, we illustrate them here by providing a new functorial description of certain integral homology groups of the Eilenberg–Mac Lane spaces $K\left(A,n\right)$ for $A$ a free abelian group. In particular, we give a complete functorial description of the groups $H_{\ast }\left(K\left(A,3\right);\mathbb{Z}\right)$ for such $A$.