Cyclic homology of groups and the Bass conjecture
Cyclic homology of the rational group algebra \({\mathbb{Q}}G\) of a group G has a direct sum decomposition \[ HC_ i({\mathbb{Q}}G)=\oplus '_{[x]}[H_*(C_ x/<x>;{\mathbb{Q}})\otimes H_*({\mathbb{C}}P^{\infty};{\mathbb{Q}})]_ i\oplus \oplus ''_{[x]}H_ i(C_ x/<x>;{\mathbb{Q}}), \] where the summands are indexed by the conjugacy classes [x] of G, \(x\in G\), \(C_ x\) is the centralizer of x in G, \(\oplus '\) (resp. \(\oplus '')\) is summation over all [x] with x of finite (resp. infinite) order [\textit{D. Burghelea}, Comment. Math. Helv. 60, 354- 365 (1985; Zbl 0595.16022)]. The author shows that if G is of finite homology dimension \(hd_{{\mathbb{Q}}}G=n\) then the \(\oplus ''\)-part vanishes for \(i\geq n\) provided (a) G is soluble, or (b) \(n=2\), or (c) G is linear over a field of characteristic 0. As a consequence he obtains that for those groups G the image of the Connes character \(Ch^{\ell}: K_ 0({\mathbb{Q}}G)\to HC_{2\ell}({\mathbb{Q}}G)\), which is compatible with the Connes-Gysin map \(HC_{2\ell}({\mathbb{Q}}G)\to H_{2\ell -2}({\mathbb{Q}}G)\), is concentrated on the \(\oplus '\) summands. This applies, in particular, to \(Ch^ 0: K_ 0({\mathbb{Q}}G)\to HC_ 0({\mathbb{Q}}G)\)- which is the same as the Hattori- Stallings rank of G - and hence is a contribution towards the strong Bass-Conjecture that the Hattori-Stallings rank of a finitely generated projective \({\mathbb{Q}}G\)-module is concentrated on the conjugacy classes of elements of finite order.
finitely generated projective \({bbfQ}G\)-module
scientific article; zbMATH DE number 3991781