Derived functors of the divided power functors

Breen, Lawrence¹ ; Mikhailov, Roman² ; Touzé, Antoine³

¹ LAGA, CNRS (UMR 7539), Université Paris 13, 99, avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France
² Chebyshev Laboratory, St Petersburg State University, St Petersburg Department of Steklov Mathematical Institute, 14th Line, 29b, Saint Petersburg, 199178, Russia
³ Laboratoiré Paul Painlevé, CNRS (UMR 8524), Université Lille 1, F-59000 Lille, France

Geometry & topology, Tome 20 (2016) no. 1, pp. 257-352.

Résumé

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 $A$.

Détail

DOI : 10.2140/gt.2016.20.257

Classification : 18G55, 55P20
Keywords: strict polynomial functors, derived functors of non-additive functors, Eilenberg–Mac Lane spaces

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