Chain maps between Gruenberg resolutions

Abstract

Let G be a finite group. For a given presentation of G = hF|Ri, Gruenberg gives a construction of a projective resolution for Z as a ZG-module. This resolution, which is called Gruenberg resolution, only depends on the ideals IF := ker{ZF → Z} and J := ker{ZF → ZG} (see [1]). We write standard resolution as a Gruenberg resolution by following the construction of Gruenberg [2]. We get an explicit chain map formula between Gruenberg resolution for standard presentation and the Gruenberg resolution for the usual presentation of a cyclic group. Then we write an explicit chain map formula between any two Gruenberg resolutions. We also give some calculations with Gruenberg resolution.