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Cyclic homology and the Lie algebra homology of matrices. (English) Zbl 0565.17006
For associative algebras $$A$$ with identity over commutative rings $$k$$ A. Connes and B. L. Tsygan defined the cyclic homology $$HC_ i(A)$$, $$i\geq 0$$. The paper approaches the subject starting from a double complex suggested by Tsygan’s work. This technical tool is employed to simplify and develop the theory of Connes and Tsygan, and to give a complete proof to Theorem 6.2 announced by the authors and independently by Tsygan. There are constructed maps from cyclic homology to the de Rham cohomology, a product is introduced, and the notion of reduced cyclic homology is defined. The fifth section contains the computation of the cyclic homology for a free algebra (Proposition 5.4). For the case $$k\subseteq\mathbb Q$$ such computations were independently made by Tsygan. The main result in the last section is
Theorem 6.2: If $$k\subseteq\mathbb Q$$ then the cyclic homology is the primitive part of the homology of the Lie algebra of matrices. In particular, $$HC_{*-1}(A)$$ is an additive analogue of Quillen’s $$K$$-functors.
Theorem 6.9 is a result on stability: for $$i\leq n$$, $$H_ i({\mathfrak gl}_ n(A),k)$$ does not depend on $$n$$.
By Theorem 6.2 one can introduce on $$HC_{*-1}(A)$$ an increasing filtration $$F_*$$ where the $$n$$th term consists of those primitive classes of the homology of $${\mathfrak gl}_{\infty}$$ which are images of homology classes of $$\mathfrak{gl}_ n$$. Conjecture 6.14 is not true in general. (It is true, nevertheless, for another, decreasing filtration.) For $$F_*$$ this proposition (i.e. $$F_ n HC_{2n}(A)=0)$$ only holds in the case when $$A$$ is the coordinate ring of a smooth algebraic manifold. In general, $$A=k[V]/(V)^ 2$$, where $$\dim V=\infty$$, gives a counter-example (personal communication by B. L. Tsygan).

MSC:
 17B56 Cohomology of Lie (super)algebras 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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