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

中文翻译：

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交换环上李代数的同调

我们研究了交换环上李代数的五种不同类型的同构，这些环在场上是自然同构的。我们证明了它们在交换环上不是同构的，甚至在$\mathbb{Z}$，并研究它们之间的联系。特别是，我们证明了在李代数作为模是平坦的情况下，它们是自然同构的。作为辅助结果，我们证明了模块M在连接外部和对称幂的主理想域上的 Koszul 复形$0\to {\mathrm{\Lambda }}^n米\to 米\otimes {\mathrm{\Lambda }}^{n-1}米\to \dots \to {秒}^{n-1}米\otimes 米\to {秒}^n米\to 0$ 纯粹是非循环的。