论文标题
关于谎言代数在交换戒指上的同源性
On homology of Lie algebras over commutative rings
论文作者
论文摘要
我们研究了五种不同类型的Lie代数的同源性，这是一个自然在场上同构的交换环。我们表明，即使超过$ \ Mathbb z，$和研究之间的联系，它们也不是同构的。特别是，我们证明它们自然是同构的，而谎言代数为模块。作为辅助结果，我们证明了一个模块$ m $的Koszul复合物比主要的理想域，该域将外部和对称功率$ 0 \与λ^n m \与m \ otimeλ^{n-1} m \ to \ dots \ to \ dots \ dots \ dots \ dots as^n-1} m \ is us us usimimeλ^{n-1} m \ is us us usime无环。
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 Λ^n M\to M \otimes Λ^{n-1} M \to \dots \to S^{n-1}M \otimes M \to S^nM\to 0 $ is purely acyclic.
