TY - JOUR

T1 - On homology of Lie algebras over commutative rings

AU - Ivanov, Sergei O.

AU - Pavutnitskiy, Fedor

AU - Romanovskii, Vladislav

AU - Zaikovskii, Anatolii

N1 - Publisher Copyright: © 2021 Elsevier Inc.

PY - 2021/11/15

Y1 - 2021/11/15

N2 - 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→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 is purely acyclic.

AB - 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→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 is purely acyclic.

KW - Comonad derived functors

KW - Dold-Puppe derived functors

KW - Homology

KW - Koszul complex

KW - Lie algebra

KW - Simplicial homology

UR - http://www.scopus.com/inward/record.url?scp=85110067617&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.06.019

DO - 10.1016/j.jalgebra.2021.06.019

M3 - Article

AN - SCOPUS:85110067617

VL - 586

SP - 99

EP - 139

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -