process

Inheritance of isomorphism conjectures under colimits 47isomorphisms (see Lemma 6.2)colim i2I K n .R Ì G i / Š ! K n .R Ì G/Icolim i2I KH n .R Ì G i / Š ! KH n .R Ì G/Icolim i2I L hn 1i .R Ì G i / Š ! L h n1i .R Ì G/Icolim i2I K n .A Ì l 1 G i / Š ! K n .A Ì l 1 G/Icolim i2I K n .A Ì m G i / Š ! K n .A Ì m G/:Let A be a C -algebra with G-action by C -automorphisms. We can consider Aas a ring only. Notice that we get a commutative diagramH G n .E VCyc.G/I K A / KH n .A Ì G/Hn G.E VCyc.G/I KH A / KH n .A Ì G/ŠidH G n .E F in.G/I KH A / KH n .A Ì G/Hn G.E F in.G/I K top /A;l 1ŠHn G.E F in.G/I K topA;m /K n .A Ì l 1 G/ K n .A Ì m G/ŠH G n .E F in.G/I K topA;r / K n .A Ì r G/,where the horizontal maps are assembly maps and the vertical maps are change oftheory and rings maps or induced by the up to G-homotopy unique G-map E F in .G/ !E VCyc .G/. The second left vertical map, which is marked with Š, is bijective. Thisis shown in [4, Remark 7.4] in the case, where G acts trivially on R, the proof carriesdirectly over to the general case. The fourth and fifth vertical left arrow, which aremarked with Š, are bijective, since for a finite group H we have A Ì H D A Ì l 1H D A Ì r H D A Ì m H and hence we can apply [13, Lemma 4.6]. In particularthe Bost conjecture and the Baum–Connes conjecture together imply that the mapK n .A Ì l 1 G/ ! K n .A Ì r G/ is bijective, the map K n .A Ì l 1 G/ ! K n .A Ì m G/ issplit injective and the map K n .A Ì m G/ ! K n .A Ì r G/ is split surjective.The upshot of this discussions is: