Some remarks on Nil groups in algebraic K-theory

SOME REMARKS ON NIL GR OUPS IN ALGEBRAI C K-THEOR Y JAMES F. DA VIS Abstract. This note explains conse q uences of recent w ork of F rank Quinn for computatio ns of Nil groups in alge braic K-theor y , in particular the Nil groups occurr ing in the K-theory of polyno- mial rings, Laure nt polynomial rings, and the group ring of the infinite dihedral gr oup. 1. St a te ment of Res ul ts Let R be a r ing w ith unit. F or an integer q , let K q R b e t he algebraic K -group of Bass and Quillen. Bass defines the NK-groups N K q ( R ) = k er( K q R [ t ] → K q R ) where the map o n K -groups is induced b y the ring map R [ t ] → R , f ( t ) 7→ f (0) . The N K - groups are often called Nil-groups b ecause they a r e related to nilp otent endomorphisms of pro jectiv e R -mo dules. Let G b e a gro up. Let Or G b e its the orbit category; ob jects are G -sets G/H where H is a subgroup of G and morphisms are G - maps. Da vis-L ¨ uc k [8] define a functor K : Or G → Spectra with the ke y prop erty π q K ( G/H ) = K q ( RH ). The utility of suc h a functor is to allo w the definition of an equiv aria n t homolog y theory , indeed for a G -CW-complex X , one defines H G q ( X ; K ) = π q (map G ( − , X ) + ∧ Or G K ( − )) (see [8, section 4 and 7] for basic prop erties). Note that m a p G ( G/H , X ) = X H is the fixed p oint functor a nd that the “co efficien ts” of the homol- ogy theory are giv en by H G q ( G/H ; K ) = K q ( RH ). A family F of sub gr oups of G is a nonempty set of subgroups closed under subgroups and conjugation. F or suc h a family , E F (short for E F G ) is the classifying space for G - actions with isoto py in F . It is c haracterized up to G - homotopy t yp e as a G -CW-complex so that E H F is con tractible for subgroups H ∈ F and is empt y for subgroups H 6∈ F . Partially supp orted by a g rant from the National Science F o undation. 1 2 JAMES F. DA VIS Consider the fo llo wing families of subgroups o f G : 1 ⊂ fin ⊂ cfin ⊂ vc ⊂ a ll Here fin = { finite subgroups } cfin = { cyclic subgroups } ∪ { finite subgroups } v c = { virtually cyclic subgroups } Note that a mo del for E all G is G/G = pt, so H G q ( E all ; K ) = K q ( RG ). Note also E 1 G = E G by definition. The F arrell-Jo nes isomorphism conjecture in K -theory [10], as rein- terpreted in [8], states that H G ∗ ( E all , E v c ; K ) = 0. (Here and elsewhere, giv en a map f : A → B , w e write H ∗ ( B , A ) as a shortha nd nota t io n for H ∗ ( M ( f ) , A ) where M ( f ) is the mapping cylinder.) Theorem 1. L et M b e the set of m aximal cyclic sub gr oups o f Z n . H Z n q ( E v c , E 1 ; K ) ∼ = M C ∈M M n − 1 ≥ i ≥ 0 2  n − 1 i  N K q − i R. See Remark 8 for a desc riptio n of the isomorphism in Theorem 1. Assuming a theorem of F rank Quinn [19], prov en using con trolled top ol- ogy , w e will sho w the follo wing corollaries in Section 5. Corollary 2. K q R [ t 1 , t − 1 1 , . . . , t n , t − 1 n ] = K q R [ Z n ] ∼ = M n ≥ i ≥ 0  n i  K q − i R ⊕ M C ∈M M n − 1 ≥ i ≥ 0 2  n − 1 i  N K q − i R. The isomorphism is described explicitly in Remark 13. Corollary 3. L et q b e an inte ger. If N K j R = 0 when q ≥ j ≥ q − n + 1 , then K q R [ t 1 , . . . , t n ] = K q R. These corollaries was prov ed in [6] in the case where R is a commuta- tiv e ring con taining the rationa ls. Their tec hniques are from algebraic geometry and commutativit y is crucial for their pro of. This coro llary is a partial answ er to Bass’ question [4, Question ( I V ) n ] asking do es N K q R = 0 imply that K q R [ t 1 , t 2 ] = K q R ? Ho w- ev er, recen tly Corti ˜ nas, Haesemey er, and W eib el [6] recen tly sho w ed the answ er is no in g eneral. Corollary 15 in Section 5 is a sp ecial case of t he f ollo wing conjecture. SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 3 Conjecture 4. L et M + b e the set of m aximal cyclic sub gr oups of Z n with a gener ator having al l p ositive c o or dinates. Then N n K q R ∼ = M C ∈M + M n − 1 ≥ i ≥ 0  n − 1 i  N K q − i R. This conjecture w as prov en by Corti ˜ nas, Haesemey er, and W eib el [6, Corollary 4.2] in the case where R is a comm utativ e ring con ta ining the rationals. W e need some not a tion to stat e our next result. Giv en a ho momor- phism ϕ : G → Γ and a family F of subgroups of Γ, let ϕ ∗ F b e the smallest family of subgroups o f G con taining ϕ − 1 H for eac h H ∈ F . If α : A → A is a ring automorphism, let A α [ t ] b e the t wisted p olynomial ring and define N K q ( A, α ) = k er( K q ( A α [ t ]) → K q ( A )) . These groups we re first defined b y F arrell. Let D ∞ = Z 2 ∗ Z 2 = h a, b | a 2 = 1 = b 2 i b e the infinite dihedral group. Theorem 5. L et ϕ : Γ → D ∞ b e an epimorphis m of gr oups. L et F = k er ϕ . Cho ose ˆ t ∈ Γ so that ϕ ( ˆ t ) = ab . L et α b e the automorphism of RF given by c onjugation by ˆ t . Then H Γ q ( E ϕ ∗ cfin , E ϕ ∗ fin ; K ) ∼ = N K q ( RF , α ) . T o sa y that a group admits an epimorphism to the infinite dihedral group is equiv alent to saying that it admits a n amalgamated pro duct decomp osition G 0 ∗ F G 1 where F is of index 2 in G 0 and G 1 . Theorem 5 is applied in [7] to reduce t he F arrell-Jones Conjecture in K -theory from the fa mily of virtually cyclic gro ups to the family of finite-b y-cyclic groups. Assuming a theorem of F rank Quinn [20], w e ha ve the follow ing corollary . The notation in its statemen t will be discusse d in the next section. Corollary 6. L et ϕ : Γ → D ∞ b e an ep imorphism of gr oups and Γ = G 0 ∗ F G 1 the c orr es p ondin g amalgamate d fr e e pr o duct d e c omp osition with F = k er ϕ . Cho ose ˆ t ∈ Γ so that ϕ ( ˆ t ) = ab , and let α : RF → R F the automorphism given by c onjugation by ˆ t . Then the Waldhausen Nil gr oup is isom orphic to the F arr el l Nil gr oup: N K q ( RF ; d RG 0 , d RG 1 ) ∼ = N K q ( RF , α ) . A differen t, purely a lg ebraic pr o of of this corollary is given in [7]. 4 JAMES F. DA VIS 2. NK-groups and rela tive homology T o prov e o ur main theorems w e need a starting point. This will b e to recast the theorems of Ba ss-Heller-Swan, Bass, Quillen, F a rrell- Hsiang, and ultimately W aldhausen [23, Theorems 1 and 2] in terms of H G q ( E all , E F ; K ) for suitable families F . Let α : A → A b e a ring automorphism. W aldhausen sho ws that the maps i + : K q A α [ t ] → K q A α [ t, t − 1 ] induced b y t 7→ t i − : K q A α − 1 [ t ] → K q A α [ t, t − 1 ] induced b y t 7→ t − 1 are monomorphisms. Let N + K q ( A, α ) = i + ( N K q ( A, α )) N − K q ( A, α − 1 ) = i − ( N K q ( A, α − 1 )) W aldhausen sho ws there is a split injection i + ⊕ i − : N + K q ( A, α ) ⊕ N − K q ( A, α − 1 ) → K q ( A α [ t, t − 1 ]) and a W ang type exact sequence · · · → K q A 1 − α − − → K q A → K q A α [ t, t − 1 ] N + K q ( A, α ) ⊕ N − K q ( A, α − 1 ) → · · · Rephrasing this in terms of group theory , let Γ 0 b e a group whic h maps epimorphically to Z . Then Γ 0 = F ⋊ α Z and there is a W ang t yp e exact sequence · · · → K q ( RF ) 1 − α − − → K q ( RF ) → K q ( R Γ 0 ) N + K q ( RF , α ) ⊕ N − K q ( RF , α − 1 ) → · · · In this section w e also consider in pa rallel the case Γ = G 0 ∗ F G 1 where F is a subgroup of G 0 and G 1 , not necessarily of index 2. Let d RG i b e the RF - bimo dule R [ G i − F ] and define N K q ( RF ; d RG 0 , d RG 1 ) = f Nil q − 1 ( RF ; d RG 0 , d RG 1 ) . This reduced Nil gro up is a subgroup of the K - theory of the exac t category defined b y W aldhausen [23 ] K q Nil( RF ; d RG 0 , d RG 1 ) = f Nil q ( RF ; d RG 0 , d RG 1 ) ⊕ K q ( RF ) ⊕ K q ( RF ) When q < 1, we use the nonconnectiv e ve rsion due to Ba r tels-L ¨ uc k [1, Section 10 ] to define the K - theory o f the Nil category . W aldhausen ga ve a split injection N K q ( RF ; d RG 0 , d RG 1 ) → K q ( R Γ) SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 5 and a May er-Vietoris ty p e long exact sequence · · · → K q ( RF ) → K q ( RG 0 ) ⊕ K q ( RG 1 ) → K q ( R Γ) / N K q → · · · The main purp ose of this section is the statemen t a nd indicatio n of the pro o f of the follo wing lemma. Lemma 7. L et F 0 b e the smal les t f a mily of sub gr oups of Γ 0 c ontaining F . L et F b e the smal lest family of sub gr oups of Γ c ontaining G 0 and G 1 . (1) The fol lowing exact se quenc es ar e split, and henc e short exact H Γ 0 q ( E F 0 ; K ) → H Γ 0 q ( E all ; K ) → H Γ 0 q ( E all , E F 0 ; K ) H Γ q ( E F ; K ) → H Γ q ( E all ; K ) → H Γ q ( E all , E F ; K ) (2) These r elative terms c an b e expr esse d in terms of Nil g r oups: H Γ 0 q ( E all , E F 0 ; K ) ∼ = N K q ( RF , α ) ⊕ N K q ( RF , α − 1 ) H Γ q ( E all , E F ; K ) ∼ = N K q ( RF ; d RG 0 , d RG 1 ) Pr o of. Define E F 0 = E F 0 Γ 0 as a pushout o f Γ 0 -spaces (7.1) S 0 × Γ 0 /F Γ 0 /F D 1 × Γ 0 /F E F 0 ✲ ❄ ❄ ✲ where t he upp er horizontal “ attac hing” map takes − 1 × F 7→ F and 1 × F 7→ ˆ tF , where ˆ t ∈ Z ⊂ Γ 0 is a generator. Then E F 0 = R , E F 0 / Γ 0 = S 1 , and there is a W ang ty p e long exact seque nce · · · → K q ( RF ) 1 − α − − → K q ( RF ) → H Γ 0 q ( E F 0 ; K ) → · · · Hence all w e really need to do is to iden tify the map H Γ 0 q ( E F 0 ; K ) → H Γ 0 q ( E all ; K ) = K q ( R Γ 0 ) with t he split injection implicit in W ald- hausen’s w ork. F or a G -CW-complex X , let K % ( X ) = map G ( − , X ) + ∧ Or G K ( − ) . This is the sp ectrum whose homotopy groups ar e H G q ( X ; K ). Sinc e mor Or Γ 0 (Γ 0 /H , Γ 0 /F ) = map Γ 0 (Γ 0 /H , Γ 0 /F ) b y definition, Y oneda’s lemma allows us to iden tify K % (with G = Γ 0 ) applied to the square 6 JAMES F. DA VIS (7.1) with the comm utativ e diagram of sp ectra (7.2) K (Γ 0 /F ) ∨ K (Γ 0 /F ) K (Γ 0 /F ) I + ∧ K (Γ 0 /F ) K % ( E F 0 ) ✲ ❄ ❄ ✲ This is a pushout diagram by [8, Lemma 6.1]. W aldhausen giv es a homotopy cart esian square (7.3) Nil ( RF ) K ( RF ) K ( RF ) K ( R Γ 0 ) ✲ ❄ ❄ ✲ (see Bartels-L ¨ uck [1, Theorem 10.6] for the nonconnectiv e version) a nd a split injection (up to homoto py) K ( RF ) ∨ K ( RF ) → N il ( R Γ 0 ) with the homotop y gro ups of the cofib er b eing N K ∗ +1 ( RF , α ) ⊕ N K ∗ +1 ( RF , α − 1 ). F urthermore, W aldhausen shows that the compo site of b o undary map in the homoto py exact sequence of the square (7.3 ) with t he pro jection on the N K -groups K q ( R Γ 0 ) → N K q ( RF , α ) ⊕ N K q ( RF , α − 1 ) is a split surjection. The square (7.2) maps to the square ( 7 .3) 1 . T racing through the ab ov e (examine the low er righ t corners!) gives a split short exact se- quence 0 → H Γ 0 q ( E F 0 ; K ) → H Γ 0 q ( E all ; K ) → N K q ( RF , α ) ⊕ N K q ( RF , α − 1 ) → 0 This giv es a pro of of the Γ 0 -part of the lemma. 1 There is a subtle po int here. W aldha usen (see also [1, The o rem 10.6 ]) shows that (7.3) is homotopy ca rtesian with resp ect to a certain natural transforma tio n betw ee n the tw o functors corr esp onding to the tw o different w ays from going to the upp er left to the low er rig ht. How ever, this natural transformation inv olves the nilpo tent structure and is the identit y on the image of (7.2). SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 7 The pro of of the Γ-part of the lemma is quite similar. Here w e will only note that E F is constructed as a pushout S 0 × Γ /F Γ /G 0 ∐ Γ /G 1 D 1 × Γ /F E F ✲ ❄ ❄ ✲ and that E F = R and E F / Γ = [0 , 1 / 2].  3. Proo f of Theorem 1 Recall the statemen t of Theorem 1. Theorem 1. L et M b e the set of m aximal cyclic sub gr oups o f Z n . H Z n q ( E v c , E 1 ; K ) ∼ = M C ∈M M n − 1 ≥ i ≥ 0 2  n − 1 i  N K q − i R. Pr o of. W e mak e use of par t icular mo dels for E 1 Z n = E fin Z n and E v c Z n , ensuring that they are Z n -CW-complexes. Enum erate the maximal cyclic subgroups of Z n as M = { C 0 , C 1 , C 2 , . . . } . L et E 1 = R n × [0 , ∞ ) × I and define E v c as the pushout ∞ a j =0 R n ∞ a j =0 R n / ( C j ⊗ R ) E 1 E v c ✲ ❄ ❄ ✲ where the j -th copy of R n is identifie d with R n × { j } × { 1 } ⊂ E 1 and the Z n -actions on all spaces are induced b y the t r a nslation action of Z n on R n and the t r ivial action on [0 , ∞ ) × I . Let Z n − 1 j ⊂ Z n denote a subgroup so tha t Z n = Z n − 1 j ⊕ C j . Let R j = C j ⊗ R ⊂ R n and R n − 1 j = Z n − 1 j ⊗ R ⊂ R n . Then 8 JAMES F. DA VIS H Z n q ( E v c , E 1 ; K ) ∼ = ← − M j H Z n q ( R n / R j , R n ; K ) ∼ = M j M i H i ( R n − 1 j / Z n − 1 j ) ⊗ H C j q − i ( ∗ , R j ; K ) ∼ = M j M n − 1 ≥ i ≥ 0  n − 1 i  H Z q − i ( ∗ , R ; K ) ∼ = M j M n − 1 ≥ i ≥ 0  n − 1 i  ( N K q − i R ⊕ N K q − i R ) , where the first isomorphism f o llo ws f rom the excision and disjoint union axioms, the second follow s from the A tiy ah-Hirzebruc h Sp ectral Se- quence [8, Theorem 4.7] which collapses at E 2 , the third f rom the homology of the to rus, and t he last from Lemma 7 with Γ 0 = Z .  An a lternativ e metho d of computing H Z n q ( R n / R j , R n ; K ) is to use the metho d of pro of of Theorem 5 b elow. Remark 8. The purp ose of this remark is to mak e the map underlying the isomorphism o f Theorem 1 a s explicit as p ossible. F or a maximal cyclic subgroup C of Z n , let CSub b e the family o f subgroups o f Z n consisting of the subgroups of C . Note that E CSub Z n = E ( Z n /C ) as Z n spaces. F or an infinite cyclic group C with generator t , there is a map 2 N K j ( R ) = N j K R ⊕ N j K R → K j RC induced b y the tw o ring maps R [ t ] → RC giv en b y t 7→ t and t 7→ t − 1 . The C -comp onen t of the isomorphism in Theorem 1 is t he comp osite of the maps M i  n − 1 i  2 N K q − i R → M i  n − 1 i  K q − i RC ∼ = H q ( B ( Z n /C ); K ( RC )) = H Z n q ( E CSub ; K ) → H Z n q ( E v c ; K ) → H Z n q ( E v c , E 1 ; K ) The only map which is not explicit is the isomorphism ∼ = . T his only dep ends on a n iden tification of Z n /C with Z n − 1 and the axioms of a generalized homology theory . SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 9 Here note for that for a generalized homology theory H , t here is a canonical iden tification (8.1) H q ( S 1 ) = H q (pt) ⊕ H q − 1 (pt) and hence a similar iden tification H q ( T n ) = L  n i  H q − i (pt). The iden- tification 8.1 uses the fact that a circle has a p oint as a retract and the isomorphisms H q ( S 1 , pt) ∼ = ← − H q ( D 1 , S 0 ) ∼ = − → H q − 1 ( S 0 , {− 1 } ) ∼ = ← − H q − 1 ( { +1 } ) . Remark 9. The group Z n satisfies the prop ert y that ev ery virtually cyclic subgroup is con tained in a unique maximal virtual cyclic sub- group. F or suc h a group G , L ¨ uc k-W eiermann [16, Section 6] hav e sho wn H G ∗ ( E v c G, E fin G ; K ) ∼ = L C ∈M H C ∗ ( E v c C , E fin C ; K ) where M is a set of represen tativ es for the conjuga cy classes of maximal virtually cyclic subgroups of G . This also can b e analyzed using the tech niques of Davis -L ¨ uc k [9, Section 4]. 4. Proo f of Theorem 5 Recall the statemen t of Theorem 5. Theorem 5. L et ϕ : Γ → D ∞ b e an epimorphis m of gr oups. L et F = k er ϕ . Cho ose ˆ t ∈ Γ so that ϕ ( ˆ t ) = ab . L et α b e the automorphism of RF given by c onjugation by ˆ t . Then H Γ q ( E ϕ ∗ cfin , E ϕ ∗ fin ; K ) ∼ = N K q ( RF , α ) . Pr o of. Let ϕ : Γ → D ∞ b e an epimorphism. The infinite dihedral group D ∞ = h a, b | a 2 = 1 = b 2 i acts on R via a ( x ) = − x a nd b (1 / 2 + x ) = 1 / 2 − x as well a s o n S ∞ via a ( x ) = − x = b ( x ). Then one can c ho ose mo dels E ϕ ∗ fin Γ = E fin D ∞ = R E ϕ ∗ cfin Γ = E cfin D ∞ = S ∞ ∗ R . This join mo del w as p oin ted o ut by Ian Ham bleton. Hence H Γ q ( E ϕ ∗ cfin Γ , E ϕ ∗ fin Γ; K ) = H Γ q ( R , S ∞ ∗ R ; K ) T o compute this relativ e homology we make a categorical divers ion. F or a subgroup H of G and a family F of subgroups of G , let F ∩ H denote the family of subgroups of H giv en b y { F ∈ F : F is a subgroup of H } . 10 JAMES F. DA VIS Lemma 10. L et F b e a famil y of sub gr oups of G . L et E b e an Or G -sp e ctrum, that is, a functor E : Or G → Sp ectra . Ther e is an Or G -sp e ctrum E F and a map of Or G - Sp ectra E F → E satisfying the fol lowing two pr op erties: (1) F or any s ub gr oup H of G , o n e c an identify the cha nge of sp e ctr a map H G q ( G/H ; E F ) → H G q ( G/H ; E ) with the change of sp ac e map H H q ( E F ∩ H H ; j ∗ E ) → H H q (pt; j ∗ E ) = π q E ( H /H ) . By j ∗ E we me an the c omp osite Or H j − → Or G E − → Sp ectra wher e j ( H /K ) = G/K . (2) F or any family G c ontaining F , one c an identify the change of sp e ctr a map H G q ( E G G ; E F ) → H G q ( E G G ; E ) with the change of sp ac e map H G q ( E F G ; E ) → H G q ( E G G ; E ) . W e will prov e a generalization of this lemma lat er, but for now note that E F ( G/H ) = map H ( − , E F ∩ H H ) + ∧ Or H E ( − ) . Let ϕ : Γ → D ∞ b e a n epimorphism. Let Γ = G 0 ∗ F G 1 b e the corre- sp onding amalg amated pro duct decompo sition. Then Γ 0 = ϕ − 1 ( Z ) = F ⋊ α Z is an index 2 subgroup. Conjug a tion b y any elemen t g ∈ Γ − Γ 0 lea v es N + K q ( RF , α ) ⊕ N − K q ( RF , α − 1 ) ⊂ K q ( R Γ 0 ) in v arian t and interc hanges the t w o summands. Let N b e the Or Γ -sp ectrum giv en as the cofib er of K ϕ ∗ fin → K . (The letter N is used to remind the reader of Nil. A similar construction w as in a preprint of F rank Quinn.) F or H ∈ ϕ ∗ fin, not e E ϕ ∗ fin ∩ H H = pt, so by Lemma 10(1), π q N (Γ /H ) = 0 . By Lemma 10(1) and L emma 7, π q N (Γ / Γ 0 ) = H Γ q ( E all Γ , E ϕ ∗ fin Γ; K ) = N K q ( RF , α ) ⊕ N K q ( RF , α − 1 ) SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 11 No w finally w e can pro v e Theorem 5. H Γ q ( E ϕ ∗ cfin Γ , E ϕ ∗ fin Γ; K ) = H Γ q ( E ϕ ∗ cfin Γ; N ) = H Γ q ( S ∞ ∗ R ; N ) = H Γ q ( S ∞ ; N ) = H 0 ( Z 2 ; N K q ( RF , α ) ⊕ N K q ( RF , α − 1 )) = N K q ( RF , α ) W e lab el the Equalities 1, 2, 3, 4, and 5. Equalit y 1 follows from Lemma 10(2). Equalit y 2 follows from the giv en mo del of the classifying space. F or x ∈ S ∞ ∗ R − R , the isotropy s ubgro up Γ x ∈ ϕ ∗ fin Γ, so π q N (Γ / Γ x ) = 0. Thu s the E 2 -terms of the sp ectral sequences o f the sk eleta fitr a tions (see [8, Theorem 4.7]) conv erging to H Γ q ( S ∞ ; N ) and H Γ q ( S ∞ ∗ R ; N ) agree, so Equality 3 follow s. In fact, since the isotropy is constan t, the E 2 -term is E 2 i,j = H i ( R P ∞ ; N K j ( RF , α ) ⊕ N K j ( RF , α − 1 )) , and computing with lo cal co efficien ts, Equalities 4 and 5 f ollo w. This completes the pro of of Theorem 5.  4.0.1. A c ate goric al diversion. W e need t o review some of the material in [8] to state and prov e the next lemma and deduce Lemma 10 . Let C b e a category . A C -sp ac e is a functor X : C → T op; a C -sp e ctrum is a functor E : C → Sp ectra. A map of C -spaces or C -sp ectra is a natural transformation. A homotopy of maps of C - spaces is a ma p X × I → Y . It is then clear what a homotopy equiv a lence of C -spaces is. A we ak homotopy e quivalenc e X → Y of C - spaces is a map whic h induces a w eak homotopy equiv a lence X ( c ) → Y ( c ) for all o b jects c of C . A C - CW-complex is a C -space X t o gether with a filtration X 0 ⊂ X 1 ⊂ X 2 ⊂ · · · ⊂ X satisfying certain prop erties; the precise definition is giv en in [8]. F or example, if Y is a G - CW-complex, then map G ( − , Y ) is a Or G -CW- complex. A C -CW appr oximation is a weak ho mo t o p y equiv alence X ′ → X where X ′ is a C -CW-complex. C -CW-approximations exist and are unique up to homotopy . Let X : C op → T op b e a C op -space and E : C → Sp ectra b e a C -sp ectrum. One can form the balanced pro duct X + ∧ C E ; 12 JAMES F. DA VIS this is a sp ectrum. Let X ′ → X b e a C op -CW-appro ximation. One defines H C q ( X ; E ) = π q ( X ′ + ∧ C E ) . This generalized homolog y theory satisfies excision, is inv ariant under w eak homotopy equiv a lence, has “ co efficien ts” H C q (mor C ( − , c ); E ) = π q ( E ( c )) and satisfies H C q ( ∗ ; E ) = π q (ho colim C E ) , where ∗ denotes a C -space so that ∗ ( c ) is a p oin t for all ob jects c . T o explicate this last p oint, recall there is a C op -CW-appro ximation E C → ∗ , functorial in C . Let B C b e the classifying space of a category C ; it is the geometric realization of the simplicial set N • C whose p - simplicies are sequences of comp osable morphisms c 0 → c 1 → · · · → c p . Fixing a ob j ect c of C , define t he under c ate gory c ↓C . An ob ject in c ↓C is a morphism φ ′ : c → c ′ in C . A morphism f f r om φ ′ : c → c ′ to φ ′′ : c → c ′′ is a morphism f : c ′ → c ′′ satisfying f ◦ φ ′ = φ ′′ . Then define the bar resolution mo del E C ( c ) = B ( c ↓C ). Note E C + ∧ C E is the usual definition of ho colim C E . Let F : B → C b e a functor. F or a C -space X , define a B -space F ∗ X ( b ) = X ( F ( b )). F or a B - space X , define a C -space F ∗ X ( c ) = mor C ( F ( − ) , c ) × B X ( − ) . There are similar definitions for sp ectra. Thes e constructions satisfy n umerous adjoin t pro p erties. If X is a B op -space and E is a C -sp ectrum, there is a homeomorphism of sp ectra X + ∧ B F ∗ E ∼ = ( F ∗ X ) + ∧ C E , natural in X and E . Similarly , if X is a B -space and Y is a C -space map B ( X , F ∗ Y ) ∼ = map C ( F ∗ X , Y ) , natural in X and Y . Next w e mo ve on to assem bly maps. The functor ( b ↓B ) → ( F ( b ) ↓C ) , ( φ : b → b ′ ) 7→ ( F ( φ ) : F ( b ) → F ( b ′ )) induces a map of B -spaces E B → F ∗ E C , and hence, b y the ab o ve adjoin t prop ert y , a map of C -spaces F ∗ E B → E C . W e call this the F -pr e-assembly map . The comp osite E B + ∧ B F ∗ E ∼ = F ∗ E B + ∧ C E → E C + ∧ C E , SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 13 as w ell as the induced map on homotop y groups H B q ( ∗ ; F ∗ E ) → H C q ( ∗ ; E ) , is called the ( F , E ) -assembly map . W e need some mor e nota tion for the statemen t and pro of of the follo wing lemma. Let F : B → C and E : C → Sp ectra b e f unctor s. F or an ob ject c of C , define the o v ercategory F ↓ c , whose ob jects are pairs ( b, φ : F ( b ) → c ). There is a commutativ e dia g ram of f unctors F ↓ c C ↓ c B C Sp ectra ✲ F c ❄ P c ❅ ❅ ❅ ❅ ❅ ❘ ∆ c ❄ Q c ✲ F ✲ E where P c ( b, φ ) = b F c ( b, φ ) = φ Q c ( φ : c ′ → c ) = c ′ . Lemma 11. L et F : B → C and E : C → Sp ectra b e f unctors. Ther e is a C -sp e ctrum E F : C → Sp ectra with E F ( c ) = E ( F ↓ c ) + ∧ F ↓ c ∆ ∗ c E and a map of C -sp e ctr a E F → E satisfying the fol lowing pr op e rties: (1) F or al l obje cts c , the ma p π q E F ( c ) → π q E ( c ) c an b e id e ntifie d with the ( F c , Q ∗ c E ) -assembly map H F ↓ c q ( ∗ ; ∆ ∗ c E ) → H C ↓ c q ( ∗ ; Q ∗ c E ) . (2) F or any C -sp ac e X , ther e is an iso m orphism H C q ( X ; E F ) ∼ = H B q ( F ∗ X ; F ∗ E ) , natur al in X and E . (3) The change of sp e ctrum map H C q ( ∗ ; E F ) → H C q ( ∗ ; E ) c an b e identifie d with the ( F , E ) -assembly ma p H B q ( ∗ ; F ∗ E ) → H C q ( ∗ ; E ) . Pr o of of L emma 10 assuming L emma 1 1 . Let F b e a family o f sub- groups of G . Let Or( G, F ) b e the r estricte d orbit c ate gory ; ob jects are G -sets G/H with H ∈ F and morphisms ar e G -maps. Note that map G ( − , E F G ) is a mo del for E Or( G, F ). 14 JAMES F. DA VIS Lemma 10(1) fo llows immediately from a pplying Lemma 11(1) to the inclusion functor F : Or( G, F ) → Or( G ) and setting E F = E F , after noting the identification of categories Or( H , F ∩ H ) = F ↓ ( G/H ) G/K 7→ ( G/K → G/H , γ K 7→ γ H ) F or families F ⊂ G of subgroups of G , let F and I b e the inclusion functors Or( G, F ) F − → Or( G, G ) I − → Or G. Lemma 10( 2 ) f ollo ws from applying Lemma 11(3) to the functor F and setting E F = E I ◦ F . W e w on’t explicitly carry out the straigh tforward but tedious iden tifications of the maps in Lemma 11 ( 3) with the maps in Lemma 10(2), but will mention tw o iden tifications used: π q (map G ( − , E G G ) + ∧ Or( G, G ) I ∗ E ) = π q (map G ( − , E G G ) + ∧ Or G I ∗ E ) ( I ∗ E ) F = I ∗ ( E I ◦ F )  Pr o of of L emma 11. Define E F ( c ) = E ( F ↓ c ) + ∧ F ↓ c ∆ ∗ c E . F or a mor- phism f : c → c ′ the map E F ( f ) : E F ( c ) → E F ( c ′ ) is g iv en by the (( F ↓ f ) : F ↓ c → F ↓ c ′ , ∆ ∗ c ′ E )-assem bly map. This defines E F . The map E F ( c ) → E ( c ) is giv en b y the comp osite of the ( F c , Q ∗ c E )- assem bly map E ( F ↓ c ) + ∧ F ↓ c ∆ ∗ c E → E ( C ↓ c ) + ∧ C ↓ c Q ∗ c E and the homotopy equiv alence E ( C ↓ c ) + ∧ C ↓ c Q ∗ c E ∼ − → mor C ↓ c ( − , Id c ) ∧ C ↓ c Q ∗ c E = Q ∗ c ( E )(Id c ) = E ( c ) whic h happ ens since Id c is a final ob ject of C ↓ c . This defines the map E F → E and justifies (1). W e will next pro v e (2). There is a map of ( F ↓ c )-spaces E ( F ↓ c ) → P ∗ c mor C ( F ( − ) , c ) x ∈ E ( F ↓ c )( b, F ( b ) → c ) 7→ ( F ( b ) → c ) ∈ P ∗ c mor C ( F ( − ) , c )( b, F ( b ) → c ) = mor C ( F ( b ) , c ) The adjoin t of this map is a map P c ∗ E ( F ↓ c ) → mor C ( F ( − ) , c ) , whic h, according to [9 , p. 91], is a w eak homotop y equiv alence of B op - spaces. SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 15 T o pro v e (2), w e may ass ume that X is a C op -CW-complex, since b oth sides of (2) are in v arian t under w eak homoto p y equiv alence . According to Lemma 3.5 o f [9], the domain of X × C P c ∗ E ( F ↓ c ) → X × C mor C ( F ( − ) , c ) = F ∗ X has the homotop y t yp e of a B op -CW-complex. Hence the ab o v e map is a homotopy B op -CW-appro ximation. Th us H B q ( F ∗ X ; F ∗ E ) = π q (( X × C P c ∗ E ( F ↓ c )) + ∧ B F ∗ E ) Note X + ∧ C E F ( c ) = X + ∧ C ( E ( F ↓ c ) + ∧ F ↓ c P ∗ c F ∗ E ) = X + ∧ C ( P c ∗ E ( F ↓ c ) + ∧ B F ∗ E ) = ( X × C P c ∗ E ( F ↓ c )) + ∧ B F ∗ E (2) follows . W e will construct maps of sp ectra (11.1) E C + ∧ C ( E ( F ↓ c ) + ∧ F ↓ c ∆ ∗ c E ) E C + ∧ C E E B + ∧ B F ∗ E ❄ ✲ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ The horizon tal map is defined giv en b y the map of C -sp ectra E F → E defined ab o ve . The diagona l map is the ( F , E )-assem bly map. The v ertical homotopy equiv a lence w as essen tially defined in the pro of of (2) with X = E C : Recall E C + ∧ C ( E ( F ↓ c ) + ∧ F ↓ c ∆ ∗ c E ) = ( E C × C P c ∗ E ( F ↓ c )) + ∧ B F ∗ E Recall also E C × C P c ∗ E ( F ↓ c ) has the homotopy ty p e of a B op -CW- complex and is contractible at each ob j ect, so there exists a B op -map to E B , unique up to homotop y , whic h is a homotopy equiv alence. Then smash with F ∗ E . Mik e Mandell p o inted out that the triangle (11.1) only comm utes up to homotopy . He also indicated the pro of of homoto p y comm utativit y giv en b elo w. 16 JAMES F. DA VIS The pro of uses simplicial metho ds. In fact, the tr ia ngle (11.1) is the geometric realization of a triangle of maps of simplicial sp ectra (11.2) X • Z • Y • ❄ h ✲ f     ✒ g Let Z p = _ σ ∈N p C E (initial σ ) where initial( c 0 → · · · → c p ) = c 0 . Similarly , let Y p = _ σ ∈N p B E ( F (initial σ )) The upp er left hand corner of (11.1) is the geometric realization of a bisimplicial sp ectrum. Let N F p,q b e the bisimplicial set with elemen ts τ = ( τ 1 , τ 2 ) = ( b 0 → · · · → b p , F ( b p ) → c 0 → · · · → c q ) , sequence s of comp osable morphisms in B and C . Let X p,q = _ ( τ 1 ,τ 2 ) ∈N F p,q E ( F (initial τ 1 )) Let X • b e the diagonal simplicial set X p = X p,p . A fundamen tal fa ct [18, p. 94] is that the geometric realizations of X • and X • , • are home- omorphic. The geometric realizations of X • , Y • , and Z • are homeomorphic to the corners of the triangle (11.1). The map f in (11.2) is defined b y mapping the ( τ 1 , τ 2 ) = ( b 0 → · · · → b p , F ( b p ) → c 0 → · · · → c p ) summand of X p to the ( c 0 → · · · → c p )-summand of Z p using the map E ( F ( b 0 )) → E ( c 0 ) induced by the comp osite F ( b 0 ) → · · · → F ( b p ) → c 0 . The map h in (11.2) (using the obvious choice o f a map of B -spaces E C × C P c ∗ E ( F ↓ c ) → E B ) maps the ( τ 1 , τ 2 )-summand of X p to the τ 1 -summand of Y p using the identit y map E ( F ( b 0 )) → E ( F ( b 0 )) . SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 17 The map g in (1 1.2) maps the ( b 0 → · · · → b p )-summand o f Y p to the ( F ( b 0 ) → · · · → F ( b p ))-summand of Z p using the identit y map E ( F ( b 0 )) → E ( F ( b 0 )) . A simplicial homoto py b etw een f and g ◦ h is a sequence of maps o f sp ectra H i : X p → Z p +1 ( i = 0 , . . . , p ) satisfying certain identities (see e.g. [17, Definition 5.1]), including ∂ 0 H 0 = f and ∂ p +1 H p = g ◦ h . In our case, H i maps the ( b 0 → · · · → b p , F ( b p ) → c 0 → · · · → c p ) summand of X p to the F ( b 0 ) → · · · → F ( b i ) → c i → · · · → c p summand of Z p using the iden tit y map E ( F ( b 0 )) → E ( F ( b 0 )) for i 6 = 0 and t he comp osite induced map E ( F ( b 0 )) → E ( c 0 ) for i = 0. This provide s the desired homotop y .  5. Consequences of Quinn’s Theorems In this section w e prov e the corollaries that w e men tioned in the first section, namely Corollaries 2, 3, and 6 . T he first t w o corollaries dep end on the recen t pro o f of the isomorphism conjecture in K -theory for G = Z n : Theorem 12 (Quinn [19], 2.4 .1 and 3.3 .1 ) . H Z n ∗ ( E all , E v c ; K ) = 0 . Pr o of of Cor ol lary 2. Note t ha t for the gr o up G = Z n , E 1 = E fin = E Z n ; a useful mo del is R n with Z n acting b y t r anslations. W e sho w b elo w that exact sequence of the pair 0 → H Z n q ( E 1 ; K ) → H Z n q ( E all ; K ) → H Z n q ( E all , E 1 ; K ) → 0 18 JAMES F. DA VIS is short exact and split. Assuming this f o r no w, w e see K q R [ Z n ] = H Z n q ( E all ; K ) ∼ = H Z n q ( E 1 ; K ) ⊕ H Z n q ( E all , E 1 ; K ) = H q ( B Z n ; K ( R )) ⊕ H Z n q ( E v c , E 1 ; K ) ∼ = M i  n i  K q − i R ⊕ M i 2  n − 1 i  N K q − i R using Theorem 1 Th us we just need to sho w that the horizon tal maps H Z n q ( E 1 ; K ) H Z n q ( E all ; K ) M i  n i  K q − i R K q R [ Z n ] ❄ ∼ = ✲ ❄ ∼ = ✲ are split injectiv e. W e momen tarily write K R instead of K (so π q K R ( G/H ) = K q RH ) . W e prov e the splitting exis ts b y w ork- ing inductiv ely on n , with the inductiv e step giv en by the comp osite map H Z n q ( E all ; K R [ Z n ] ) = K q R [ Z n ] = H Z q ( E all ; K R [ Z n − 1 ] ) → H Z q ( E 1 ; K R [ Z n − 1 ] ) = K q R [ Z n − 1 ] ⊕ K q − 1 R [ Z n − 1 ] = H Z n − 1 q ( E all ; K R [ Z n − 1 ] ) ⊕ H Z n − 1 q − 1 ( E all ; K R [ Z n − 1 ] ) , where the map “ → ” is give n applying Lemma 7(1).  There is anot her decomposition of K q R [ Z n ] giv en b y the fundamental theorem of K -theory . W e will review this to compare and con trast with our decomp osition and to set the stage f o r the next corollary . T he fundamen tal t heorem of algebraic K - theory states that K q R [ t, t − 1 ] ∼ = K q R ⊕ N K q R ⊕ N K q R ⊕ K q − 1 R. More precisely , for an y functor K : Ring s → Ab elian Groups, Bass [3, Chapter XI I, Section 7 ] defines tw o functors N K ( R ) = k er( K ( R [ t ]) → K ( R )) LK ( R ) = cok( K ( R [ t ]) ⊕ K ( R [ t − 1 ]) → K ( R [ t, t − 1 ])) Bass calls K a c ontr acte d functor if the four-term sequence 0 → K ( R ) → K ( R [ t ]) ⊕ K ( R [ t − 1 ]) → K ( R [ t, t − 1 ]) → LK ( R ) → 0 SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 19 is exact and the surjection with target LK ( R ) is split, natura lly in R . He noted that N LK ( R ) ∼ = LN K ( R ) and that if K is con tra cted, then so are N K and LN . The more precise vers ion of the fundamen tal theorem of K - theory is that K q is a contracted functor ( q ∈ Z ) and there is a nat ur a l iden tification K q − 1 = LK q , whic h in fact is take n as the definition of K q for q negative. This allow s a v ery cute formulation of the calculation of the K -theory of (Lauren t) p olynomial r ing s. K q ( R [ t 1 , t − 1 1 , . . . t n , t − 1 n ]) ∼ = ( I + 2 N + L ) n K q ( R ) (12.1) K q ( R [ t 1 , . . . t n ]) ∼ = ( I + N ) n K q ( R ) (12.2) Pr o of of Cor ol lary 3. First one needs to identify the change of space map H Z n q ( E 1 ; K ) → H Z n q ( E all ; K ) with the split injection from the fundamen tal theorem of K -theory M i  n i  K q − i R = ( I + L ) n K q R → K q R [ Z n ] . This follows from the case where n = 1 whic h can b e done directly (using that the map K q − 1 R → K q R [ t, t − 1 ]) is giv en b y the pro duct with t ∈ K 1 Z [ t, t − 1 ] or by consulting [12, section 4 and 8]. Th us the isomorphism M n ≥ i ≥ 0  n i  K q − i R ⊕ M C ∈M M n − 1 ≥ i ≥ 0 2  n − 1 i  N K q − i R ∼ = ( I +2 N + L ) n K q R giv en by comparing the isomorphisms from Coro llary 2 and equation (12.1) restricts to the “identit y” on the subgroups L n ≥ i ≥ 0  n i  K q − i R → ( I + L ) n K q R . This induces an isomorphism on the quotien t M C ∈M M n − 1 ≥ i ≥ 0 2  n − 1 i  N K q − i R ∼ = (( I + 2 N + L ) n − ( I + L ) n ) K q R Th us if the left hand side v anishes N i K q R = 0 for n ≥ i ≥ 1. Hence b y equation (12.2), K q R [ t 1 , . . . t n ] = K q R as desired.  Remark 13. Since K q R [ Z n ] is, in general, a n infinitely generated ab elian g roup, it is worth b eing more explicit a b out the isomorphism in Corollary 2. T o this end, let M ( n ) b e the set of degree n monomia ls in the no n-c ommuting v ariables I and L . Let M ( n, i ) ⊂ M ( n ) b e those monomials with exactly i L ’s. F or eac h maximal infinite cyc lic subgroup C of Z n , c ho ose a basis b 1 , b 2 , . . . , b n of Z n with b 1 ∈ C , i.e. c ho ose an automorphism β C : 20 JAMES F. DA VIS Z n → Z n with β C ( Z × { (0 , . . . , 0) } ) = C . W e define an in ternal direct sum decomp osition K q R [ Z n ] = M f ∈ M ( n ) f K q R ⊕ M C ∈M M f ∈ M ( n − 1) β C ∗ ( f N + K q R ⊕ f N − K q R ) . W e indicate the summands b y a n example: LI LK q R ⊂ K q R [ t 1 , t − 1 1 , t 2 , t − 1 2 , t 3 , t − 1 3 ] is giv en by ( · t 3 )inc ∗ ( · t 1 ) K q − 2 R ⊂ ( · t 3 )inc ∗ K q − 1 R [ t 1 , t − 1 1 ] ⊂ ( · t 3 ) K q R [ t 1 , t − 1 1 , t 2 , t − 1 2 ] ⊂ K q R [ t 1 , t − 1 1 , t 2 , t − 1 2 , t 3 , t − 1 3 ] The pro of o f Corolla r y 2 demonstrates the internal direct sum decom- p osition. This direct sum decompo sition may b e easier to parse if w e write it as K q R [ Z n ] = ( I + L ) n K q R ⊕ M C ∈M β C ∗ (( I + L ) n − 1 ( N + + N − ) K q R ) The M -summands dep end only on the c hoice of a splitting of the in- jection C → Z n . If f ∈ M ( n, i ) there is a canonical isomorphism f K q R ∼ = K q − i R and for f ∈ M ( n − 1 , i ), f N ± K q R ∼ = N ± K q − i R. Remark 14. Let M 6 =0 b e the set of maximal cyclic subgroups of Z n whose generators ha ve all nonzero co ordinates. Let M + b e the set o f maximal cyclic subgroups of Z n with a generator ha ving all p ositiv e co ordinates. The pro of of Corollary 3 shows that ( N + + N − ) n K q R = M C ∈M 6 =0 β C ∗ ( I + L ) n − 1 ( N + + N − ) K q R. This implies 2 n N n K q R ∼ = M C ∈M 6 =0 M i 2  n − 1 i  N K q − i R. Corollary 15. Supp ose N K q R, N K q − 1 R, . . . , N K q − n +1 R ar e al l c ount- able torsion gr oups. Then N n K q R ∼ = M C ∈M + M i  n − 1 i  N K q − i R. SOME REMAR KS ON NIL GROUPS IN A LGEBRAIC K- THEOR Y 21 Pr o of. There is a 2 n − 1 -to-1 map M 6 =0 → M + giv en by c ho osing a generator ( x 1 , . . . , x n ) for C and sending C = h ( x 1 , . . . , x n ) i ∈ M 6 =0 7→ h ( | x 1 | , . . . , | x n | ) i ∈ M + Th us 2 n N n K q R ∼ = 2 n M C ∈M + M i  n − 1 i  N K q − i R. If A and B a r e coun table torsion ab elian groups, then A ⊕ A ∼ = B ⊕ B ⇒ A ∼ = B b y Ulm’s Theorem (see [14, Exercise 32]). The corollary follo ws.  Remark 16. If R is a countable ring so that R ⊗ Q is regular noe- therian, then N K ∗ R is coun table torsion. The fa ct that they are coun table follo ws from the fact tha t the homolo gy of B GL ( R [ Z n ]) is coun table. The fact that they are torsion comes from the fact that B GL ( R [ Z n ]) → B GL (( R ⊗ Q )[ Z n ]) is a ratio nal equiv a lence, and that N K ∗ ( R ⊗ Q ) = 0. This remark is due to Chuc k W eib el. A k ey example is R = Z G for G a finite group. Remark 17. T om F arrell told the author some y ears ago that he and Lo w ell Jones knew that the isomorphism conjecture for Z n has strik- ing consequence s for N i K and presumably an ticipated Corollary 2. Nonetheless, no w that the isomorphism conjecture has b een prov ed for Z n , it is worth while to do cumen t its consequence s. Corti ˜ nas, Haesemey er, and W eib el [6] use ra dically differen t tec h- niques to analyze N i K R . Their results are b o th stronger, since t hey ha v e a more complete computation using Ho c hsc hild homology and w eak er, b ecause their r esults apply to a restricted class of rings. It w ould b e in teresting to compare the t w o approac hes. In [2 0], F ra nk Quinn uses con trolled top ology to prov e the following theorem. Holger Reic h [21] gav e a short pro of of the theorem using a theorem from [2]. Theorem 18 (Quinn) . L et Γ → D ∞ b e an epimorphism of gr oups. Then H Γ ∗ ( E all , E φ ∗ cfin ; K ) = 0 . Pr o of of Cor ol lary 6. N K q ( RF ; d RG 0 , d RG 1 ) = H Γ ( E all , E ϕ ∗ fin ; K ) = H Γ ( E ϕ ∗ cfin , E ϕ ∗ fin ; K ) = N K q ( RF , α ) 22 JAMES F. DA VIS The first equalit y holds b y Lemma 7 ( 2), t he second b y Quinn’s theorem, and the third b y Theorem 5.  Remark 19. Sev eral y ears ago the author a nd Bog dan V a jiac outlined a unpublished pro of o f Corollary 6 for q ≤ 1, using con trolled top o l- ogy . The original motiv ation for this no te w as to che ck that Quinn’s Theorem 18 is consisten t with the work o f D a vis-V a jiac. Previous par- tial results are con tained in the pap ers [11], [13 ], [5], [15]. A pro of of Corollary 6 av oiding con tro lled to p ology is g iv en in [7]. This pap er b enefited from con v ersations with T om F arrell, Christian Haesemey er, Mike Mandell, F rank Quinn, and Ch uc k W eib el. Reference s [1] Arthur Bar tels a nd W olfgang L¨ uck, “Isomor phis m conjecture for ho motopy K - theo ry and gro ups a cting o n tre e s,” J. Pur e Appl. Algebr a 205 (2 0 06), 6 6 0– 696. [2] Arthur Bartels, W olfgang L¨ uc k, and Holger Reich, “The K-theoretic F arr e ll-Jones Co njecture for hype r b olic groups ,” Invent. Math. (to app ear). ht tp:// www.arxiv.o rg/math.K T/0701 434 . [3] Hyman Bas s, Algebr aic K -the ory , W. A. Benjamin (19 68). 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