Cyclic homology and the Lie algebra homology of matrices.

*(English)*Zbl 0565.17006For 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).

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).

Reviewer: Alice Fialowski (Budapest)

##### MSC:

17B56 | Cohomology of Lie (super)algebras |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |