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
j
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)