On homology of Lie algebras over commutative rings

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 $0\to {\Lambda }^{n}M\to M\otimes {\Lambda }^{n-1}M\to \cdots \to S^{n-1}M\otimes M\to S^{n}M\to 0$ is purely acyclic.

References in corpus (1)

• Quandle cohomology is a Quillen cohomology