Transporting cohomology in Lazard correspondence

TL;DR

This paper demonstrates that for certain degrees, the cohomology functors of finitely generated nilpotent pro-p groups and their corresponding Lie algebras are naturally equivalent under Lazard correspondence, extending to relative cohomology.

Contribution

It establishes the natural equivalence of group and Lie cohomology functors for specific degrees within Lazard correspondence, including relative cohomology groups.

Findings

Cohomology functors are equivalent for i=0,1,2,3 in the specified categories.

Results extend to relative cohomology groups.

Provides a bridge between group and Lie algebra cohomology in this context.

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 $\mathbb{Z}_p$-Lie algebras of nilpotency class smaller than $p$. Denote by $H_{Gr}^i$ and $H_{Lie}^i$ the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for $i=0$, $1$ and $1$, and for a given category of modules the cohomology functors $H_{Gr}^i\circ \textbf{exp}$ and $H^i_{Lie}$ are naturally equivalent. A similar result is proven for $i=3$ and the relative cohomology groups.