On homology of Lie algebras over commutative rings

[摘要] 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 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 -> A(n )M -> M circle times A(n-1) M -> ... -> Sn-1 M circle times M -> S-n M -> 0 is purely acyclic. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-15

[关键词] Homology;Lie algebra;Dold-Puppe derived functors;Koszul complex;Comonad derived functors;Simplicial homology