Cyclic homology of $H$-unital (pro-) algebras, Lie algebra homology of matrices, and a paper of Hanlon's

Details Cyclic homology of $H$-unital (pro-) algebras, Lie algebra homology of matrices, and a paper of Hanlon's Cyclic homology of $H$-unital (pro-) algebras, Lie algebra homology of matrices, and a paper of Hanlon's

2005-04-07

arxiv.org/abs/math/0504148v1

Mathematics

K-Theory and Homology

amslatex, 12 pages

Scientific paper

We consider algebras over a field $k$ of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces \[ H(\gl(A))\iso\wedge (HC(A)[-1]) \] between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the $k$-algebra $A$, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen and independently by Tsygan. For $H$-unital algebras, it is known to hold too, as is that the proof follows from results of Hanlon's. However, to our knowledge, the proof is not immediate, and has not been published. In this paper we fill this gap in the literature by offering a detailed proof. Moreover we establish the isomorphism in the general setting of ($H$-unital) pro-algebras.

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