process

62 A. Bartels, S. Echterhoff, and W. LückProof. We begin with H ‹. I K R/ and H ‹. I KH R/. We have to show for everydirected systems of groups fG i j i 2 I g with G D colim i2I G i together with a mapW G ! that the canonical mapscolim i2I K n .R Ì G i / ! K n .R Ì G/Icolim i2I KH n .R Ì G i / ! KH n .R Ì G/;are bijective for all n 2 Z. Obviously R Ì G is the colimit of rings colim i2I R Ì G i .Now the claim follows for K n .R Ì G/ for n 0 from [36, (12) on page 20].Using the Bass–Heller–Swan decomposition one gets the results for K n .R Ì G/ forall n 2 Z and that the mapcolim i2I N p K n .R Ì G i / ! N p K n .R Ì G/is bijective for all n 2 Z and all p 2 Z;p 1 for the Nil-groups N p K n .RG/ definedby Bass [8, XII]. Now the claim for homotopy K-theory follows from the spectralsequence due to Weibel [41, Theorem 1.3].Next we treat H ‹. I Lh 1iR/. We have to show for every directed systems of groupsfG i j i 2 I g with G D colim i2I G i together with a map W G ! that the canonicalmapcolim i2I L hn 1i .R Ì G i / ! L h n 1i .R G/is bijective for all n 2 Z. Recall from [37, Definition 17.1 and Definition 17.7] thatL hn 1i .R Ì G/ D colim m!1 Lnh miL h mi.R Ì G/In .R Ì G/ D coker L h mC1inC1.R Ì G/ ! L h mC1inC1.R Ì GŒZ/ for m 0.Since L h1in .R Ì G/ is L h n .R Ì G/, it suffices to show that! n W colim i2I L h n .R Ì G i/ ! L h n .R Ì G/is bijective for all n 2 Z. We give the proof of surjectivity for n D 0 only, the proofsof injectivity for n D 0 and of bijectivity for the other values of n are similar.The ring R Ì G is the colimit of rings colim i2I R Ì G i . Let i W R Ì G i ! R Ì Gand i;j W R Ì G i ! R Ì G j for i;j 2 I;i j be the structure maps. One candefine R Ì G as the quotient of `i2I R Ì G i= , where x 2 R Ì G i and y 2 R Ì G jsatisfy x y if and only if i;k .x/ D j;k .y/ holds for some k 2 I with i;j k.The addition and multiplication is given by adding and multiplying representativesbelonging to the same R Ì G i . Let M.m;nI R Ì G/ be the set of .m; n/-matrices withentries in R Ì G. GivenA i 2 M.m;nI R Ì G i /, define i;j .A i / 2 M.m;nI R Ì G j /and i .A i / 2 M.m;nI R Ì G/ by applying i;j and i to each entry of the matrix A i .We need the following key properties which follow directly from inspecting the modelfor the colimit above:(i) Given A 2 M.m;nI R Ì G/, there exists i 2 I and A i 2 M.m;nI R Ì G i / withi.A i / D A.