Cohomology of solvable saturable pro-$p$ groups and Lie algebras

TL;DR

This paper establishes an isomorphism between the mod-$p$ cohomology of solvable saturable pro-$p$ groups and their associated Lie algebras, revealing a deep connection between group cohomology and Lie algebra structures.

Contribution

It proves that the mod-$p$ cohomology of solvable saturable pro-$p$ groups matches that of their Lie algebras, extending understanding of their algebraic properties.

Findings

Isomorphism between group and Lie algebra cohomology for solvable saturable pro-$p$ groups.

Cohomology of the Lie algebra and its reduction modulo p are isomorphic.

Results hold for odd primes p and integer n.

Abstract

Let $p$ be an odd prime, and let $n\in \N$ be an integer. We show that the $n^{\text{th}}$ mod-$p$ cohomology of a solvable saturable pro-$p$ group is isomorphic to the $n^{\text{th}}$ mod-$p$ cohomology of its associated $\Z_p$-Lie algebra $\g$ as a $\F_p$-vector space. Addittonally, we obtain that the $n^{\text{th}}$ mod-$p$ cohomology of $\g$ and of $\g/p\g$ are isomorphic as $\F_p$-vector spaces.