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The cyclic homology of the group rings. (English) Zbl 0595.16022
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 $$HC_*(k[G\times {\mathbb{Z}}])$$.