Polynomial functors

Abstract

1. Introduction: If G is a group, Z(G) its integral group-ring and A_G the augmentation ideal, then we can form the Abelian groups P_n(G)=A_G/A_Gⁿ⁺¹ and Q_n(G)=A_Gⁿ/A_Gⁿ⁺¹. In (5) we have studied the structure of these Abelian groups which we called polynomial grouups. If C denotes the category of Abelian groups, then P_n and Q_n are functors from C into C. We call these functors polynomial functors. The object of this work is to study the nature of these funtors. Except for n = 1, these functors are non-additive. In fact, in the sense of Eilenberg-Maclane (4) these are functors of degree exactly n (Theorem 2.3). Because of their non-additive nature, their derived functors cannot be calculated in the traditional Cartan-Eilenberg(1) method. We have to make use of the more recent theory of Dold-Puppe (3).