an:03900990 Zbl 0565.17006 Loday, Jean-Louis; Quillen, Daniel Cyclic homology and the Lie algebra homology of matrices EN Comment. Math. Helv. 59, 565-591 (1984). 00149146 1984
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17B56 16E40 Hochschild homology; cyclic homology; Lie algebra of matrices; filtration; primitive classes 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). Alice Fialowski (Budapest)