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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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