Transporting cohomology in Lazard Correspondence

Abstract

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-p groups of nilpotency class smaller than p and finitely generated nilpotent Z_p-Lie algebras of nilpotency class smaller than p. Denote by H^i_{Gr} and H^i_{Lie} the group cohomology functors and the Lie cohomology functors, respectively. The aim of this paper is to show that for i=0, 1 and 2, and for a given category of modules the cohomology functors H^i_{Gr} o exp and H^i_{Lie} are naturally equivalent. A similar result is proved for i=3 with the relative cohomology groups.