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Register a new userOn homology of Lie algebras over commutative rings
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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 $0to Lambda^n Mto M otimes Lambda^{n-1} M to dots to S^{n-1}M otimes M to S^nMto 0 $ is purely acyclic.
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We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
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