The cyclic homology of the group rings. (English) Zbl 0595.16022

Comment. Math. Helv. 60, 354-365 (1985); erratum ibid. 98, No. 3, 641-642 (2023).
Let G be a discrete group. For \(x\in G\), denote by \(G_ x\) the centralizer of x and by \(N_ x\) the quotient of \(G_ x\) by the subgroup generated by x. Write \(<G>=<G>_ f\cup <G>_{\infty}\) for the decomposition of the conjugacy classes of G into classes \(\bar x\) with the property that ord x\(<\infty\), resp. ord x\(=\infty\). Main result: \[ 1)\quad HH_*(k[G])=\oplus_{\bar x\in <G>}H_*(BG_ x;k), \]
\[ 2)\quad HC_*(k[G])=\oplus_{\bar x\in <G>_ f}H_*(BN_ x;k)\otimes H_*(BS^ 1;k)+\oplus_{x\bar {\;}\in <G>_{\infty}}H_*(BN_ x;k). \] Here k is a field of characteristic zero and \(HH_*\) resp. \(HC_*\) denote the Hochschild resp. cyclic homology groups, see A. Connes [C. R. Acad. Sci., Paris, Sér. I 296, 953-958 (1983; Zbl 0534.18009)]. This nice computation, which indicates the different types of contribution to \(HC_*\) of classes of finite resp. infinite order, is carried out by identifying the whole Connes-Gysin sequence of k[G] with the one obtained starting from a certain cyclic set associated with G and then noticing a splitting (as a cyclic set) of this construction, which is parametrized by \(<G>\). The technical device is the notion of a cyclic groupoid. Along the way, similar results are derived for \(k=arbitrary\) commutative ring. As an application, Künneth-type formulae are offered for \(HC_*(k[G*H])\) and \(HC_*(k[G\times {\mathbb{Z}}])\).
Reviewer: St.Papadima


16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S34 Group rings
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
20J05 Homological methods in group theory


Zbl 0534.18009
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