Quelques propriétés homologiques des modules précroisés. (Some homological properties of precrossed modules) (Q912203)

A precrossed module consists of a homomorphism of groups \(\partial: M\to P\) together with a left action of P on M such that \(\partial (^ pm)=p(\partial m)p^{-1}\). In a precrossed module the Peiffer commutator is an element of the form \(mm'm^{-1}(^{\partial m}m')^{-1}\), comparing the action of M on itself by conjugation, with that via \(\partial\) and the action of P on M. One can construct a Peiffer series, analogous to the lower central series but replacing the rôle of the commutator by Peiffer commutators. The first main result of this paper is a precrossed module version of Stalling's result on homology and lower central series. The second main result identifies a second homology or Schur multiplier type group for precrossed modules with the kernel of a commutator map on an exterior product.