On homology of Lie algebras over commutative rings

TL;DR

This paper investigates different types of homology for Lie algebras over commutative rings, revealing their non-isomorphism in general and establishing conditions for their equivalence, with auxiliary results on Koszul complexes.

Contribution

It demonstrates that homology types are not generally isomorphic over rings and identifies conditions for their isomorphism, also proving a new acyclicity result for Koszul complexes over PIDs.

Findings

Homology types are not isomorphic over commutative rings.

Homology types are isomorphic for flat Lie algebras.

Koszul complex over PIDs is purely acyclic.

Abstract

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $\mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0\to \Lambda^n M\to M \otimes \Lambda^{n-1} M \to \dots \to S^{n-1}M \otimes M \to S^nM\to 0 $ is purely acyclic.