On the Weak Homological Dimension of the Group Algebra of Solvable Groups

Abstract

Let KG denote the group algebra of the group G over a field K of characteristic 0. The weak (homological) dimension of KG is defined as the largest integer q for which there exist KG-modules B, C such that Torq G(B, C) 0. By [l; p. 352] q is equal to the largest integer for which there exists a KG-module A such that Hq(G, A) 0.We write w.dim KG = q. The following is a corollary of the main result (Theorem 1): For a solvable group G the weak dimension of KG is equal to the Hirsch number hG of G, w.dim KG = hG.As an application we obtain (Theorem 5) : If G is a nilpotent group of finite Hirsch number hG, then hG is equal to the largest integer q for which Hq (G, K + ) 0 0, where K + denotes the additive group of K with trivial G-operation.The starting point of our investigation was a result of K. W. Gruenberg [3], communicated privately to the author, which relates the Hirsch number of a torsionfree nilpotent group to the cohomological dimension of the group. We would like to express our sincere thanks to Gruenberg for showing us his result. Definition. Let G be a group in C, and {Ni} a series of normal subgroups with the above property. We define the Hirsch number hG of G to be the sum of the ranks of those quotients N^/N i+ 1 which are abelian. If this sum is not finite, we write hG = oo. The Hirsch number of G and the weak dimension of KGThe fact that hG does not depend on the series of subgroups is well known; also, it follows immediately from our Theorem 1. (In the literature the Hirsch number is usually defined for a slightly different class of groups, see for example [4; p. 150]. However, the above definition is better for our purpose.) THEOREM 1 . For G in C we have hG = w.dim KG.We first prove a special case: PxorosiTtorr 2 [2]. Let N be an abelian group of rank n. Then w.dim KN = n and H^(N, K + ) = K.Proof. We proceed by induction on n.

Harmonic analysis, cohomology, and the large-scale geometry of amenable groups

“…Recall that by a result of Stammbach [51], for every solvable G (not necessarily torsion free) hG=hdQ G, and that for every countable group G and a ring R one has hdRG~cdRG~hdRG+I (cf. [9,Theorem 4.6]).…”

On dimensions in Bredon homology

“…We write hG for the Hirsch length of G. Proof. Obviously, from 4.3 and Stammbach's result [23] it follows that hdG hG. We need to show the other inequality.…”

“…[20], page 132. Since H is torsion-free, Stammbach's result [23] implies that hH = hdH = hdH. We now use a similar argument as in Theorem 5.5, but this time we apply it toF, the completion of F. Now H = {τ (G)}, and the families H andF satisfy the conditions of [16, 4.5] and hence hd H G = hdFG.…”

“…Here we show it for nilpotent groups using a purely algebraic approach following ideas of Stammbach [23] and Bieri [1], who related the (co)homological dimension of soluble groups to the Hirsch length hG. For torsion-free soluble groups a result of Stammbach [23] implies that hdG = hG, and, if in addition G is countable, then hG cdG hG + 1. For a detailed account of these facts see [1].…”

Groups which satisfy a weak form of Poincaré duality

“…Our construction of <2) G was based on that of the module A used in Stammbach's proof [11] that the homological dimension of a group G in the class C is equal to the Hirsch length hG, where C is composed of the groups whose factors are locally finite or abelian. C £ # , and for GeC, A = S> G and n G =hG where ria^Y,^1) i n t h e notation of (2).…”