process

Inheritance of isomorphism conjectures under colimits 43Our formulation of these conjectures follows the homotopy theoretic approach in[13]. The original assembly maps are defined differently. We do not give the proofthat our maps agree with the original ones but at least refer to [13, page 239], wherethe Farrell–Jones conjecture is treated and to Hambleton–Pedersen [21], where suchidentification is given for the Baum–Connes conjecture with trivial coefficients.1.4 Inheritance under colimits. The main purpose of this paper is to prove thatthese conjectures are inherited under colimits over directed systems of groups (with notnecessarily injective structure maps). We want to show:Theorem 1.8 (Inheritance under colimits). Let fG i j i 2 I g be a directed systemof groups with (not necessarily injective) structure maps i;j W G i ! G j . Let G Dcolim i2I G i be its colimit with structure maps i W G i ! G. Let R be a ring (withinvolution) and let A be a C -algebra with structure preserving G-action. Given i 2 Iand a subgroup H G i , we let H act on R and A by restriction with the grouphomomorphism . i /j H W H ! G. Fixn 2 Z. Then:(i) If the assembly mapasmb H n W H H n .E VCyc.H /I K R / ! H H n .fgI K R/ D K n .R Ì H/of (1.1) is bijective for all n 2 Z, all i 2 I and all subgroups H G i , then forevery subgroup K G of G the assembly mapasmb K n W H K n .E VCyc.K/I K R / ! H K n .fgI K R/ D K n .R Ì K/of (1.1) is bijective for all n 2 Z.The corresponding version is true for the assembly maps given in (1.2), (1.3),(1.4), and (1.6).(ii) Suppose that all structure maps i;j are injective and that the assembly mapasmb G inW H G in .E VCyc.G i /I K R / ! H G in .fgI K R/ D K n .R Ì G i /of (1.1) is bijective for all n 2 Z and i 2 I . Then the assembly mapasmb G n W H G n .E VCyc.G/I K R / ! H G n .fgI K R/ D K n .R Ì G/of (1.1) is bijective for all n 2 Z.The corresponding statement is true for the assembly maps given in (1.2), (1.3),(1.4), (1.5), and (1.6).Theorem 1.8 will follow from Theorem 4.5 and Lemma 6.2 as soon as we haveproved Theorem 6.1. Notice that the version (1.5) does not appear in assertion (i). Acounterexample will be discussed below. The (fibered) version of Theorem 1.8 (i) inthe case of algebraic K-theory and L-theory with coefficients in Z with trivial G-actionhas been proved by Farrell–Linnell [16, Theorem 7.1].