On homology of linear groups over k[t]

ON HOMOLOGY OF LINEAR GROUP S OVER k [ t ] MA TTHIAS WENDT Abstract. This note explains how to prov e that for any simply-connected reductiv e group G and any infinite field k , the inclusion k ֒ → k [ t ] induces an isomorphism on group homology . This generalizes r esults of Soul´ e and Knu dson. Contents 1. Int ro duction 1 2. Preliminary reduction 2 3. Bruhat-Tits bu ildings and the So ul ´ e-Marga ux-theorem 3 4. Homology o f the stabilizers and Kn udson’s theorem 5 References 10 1. Introduction The question of ho motopy in v ariance of group homolo gy is the questio n under which conditions on a linea r algebraic g roup G and a comm utativ e r ing R the natural morphism G ( R ) → G ( R [ t ]) induces isomorphisms in g roup homolog y . This is an unstable version of homotopy in v aria nce for algebraic K -theory as esta blished by Quillen in [Qui73]. The t wo main results whic h have b een obtained in this direction a re due to Soul´ e and Knudson. In [Sou7 9], Soul´ e determined a fundamental doma in for the action of G ( k [ t ]) on the a sso ciated B ruhat-Tits building and deduced homotopy inv ariance for fields of characteristic p > 0 with field co efficients prime to p . In [Knu97], Knudson extended Soul ´ e’s appr oach and deduced ho mo topy inv ariance with integral co efficients for S L n ov er infinite fields. In this pap er, we gener alize Knudson’s theor em to ar bitrary (connected) reduc- tive groups, cf. Theorem 4.7: Theorem 1.1. L et k b e an infinite field and let G b e a c onne cte d r e ductive smo oth line ar algebr aic gr oup over k . Th en t he c anonic al inclusion k ֒ → k [ t ] induc es iso- morphisms H • ( G ( k ) , Z ) ∼ = − → H • ( G ( k [ t ]) , Z ) , if the or der of t he fundamental gr oup of G is invertible in k . It follo ws from the w o rk of Krsti´ c and McCoo l [KM97] that homotopy inv ariance do es not work for H 1 of ra nk one groups o ver integral domains which are not fields. In the case of rank tw o groups, homotopy inv ariance fails for H 2 as discusse d in [W en12]. It therefore seems that one cannot hop e for an extension of the ab ov e result for arbitra ry regular rings o r even p olynomial ring s in more than 1 v aria ble. 1991 M athematics Subje ct Classific ation. 20G10,20E42. Key wor ds and phr ases. l inear gr oups, p olynomial r ings, group homology , homotop y inv ariance. 1 2 MA TTHIAS WENDT Structure of the p ap er: In Section 2, we reduce to simply-connected absolutely almost s imple g roups. Section 3 reca lls the necessa ry facts on Bruhat-Tits theory and Margaux’s generalizatio n of Soul´ e’s theorem. In Section 4, we e x tend the homology computations of K nudson to groups other than S L n . A cknow l e dgements: I have to ap olog ize for the numerous gaps in previous ver- sions of this work. I would like to thank Mike Hopkins for p ointing o ut a critical error in an ear lier version, and Chr istian Haes emeyer a nd Aravind Asok for p oint- ing out proble ms with Knud son’s injectivit y argument in the rank tw o ca se. I also thank Ar avind Asok, Christian Haesemeyer, Roy Joshua and F abien Mo rel for many useful co mmen ts and their interest in the homotopy inv ariance q ue s tion. 2. Preliminar y reduction In this s ection, we provide so me pr eliminary reductions. More precisely , Theorem 1.1 follows for all reductive groups if it can b e shown for simply-connected almost sim- ple groups. The arguments are fairly standard reductions, ubiquitous in the theory of alg ebraic gr oups. Recall fro m [Bor9 1 ] the basic notio ns of the theor y of linea r alg ebraic groups. In particular, the r adic al R ( G ) o f a group G is the largest co nnected s olv able normal subgroup of G , and the unip otent r adic al R u ( G ) of G is the la rgest connected unipo ten t no rmal subgroup of G . A connected group G is ca lle d r e ductive if its unipo ten t r adical is trivial, and it is ca lled semisimple if its r adical is trivial. A connected alg ebraic g roup G is c a lled simple if it is non-commutativ e and has no nontrivial no rmal a lgebraic subgro ups. It is c alled almost simple if its centre Z is finite and the quotient G/ Z is simple. A semisimple group is c a lled simp ly- c onne cte d , if there is no no ntrivial isog eny φ : ˜ G → G . The a dditiv e gro up is denoted by G a , and the multiplicativ e gr oup by G m . A torus is a linea r algebra ic g roup T defined ov er a field k which over the algebr aic closure k is iso morphic to G n m for s ome n . W e now show that the main theore m follows for r eductive gro ups if it ca n b e prov ed for a lmost simple simply-co nnected g roups. W e first r educe to semisimple groups, the ba sic idea to keep in mind is the seque nce S L n → GL n → G m which reduces homotopy in v ariance for GL n to S L n . Prop osition 2.1. T o pr ove The or em 1.1, it su ffic es t o c onsider t he c ase wher e G is semisimple over k . Pr o of. F o r a reductive group G , we hav e a split extension of linear algebra ic g roups 1 → ( G, G ) → G → G/ ( G, G ) → 1 where ( G, G ) denotes the comm utator subgro up in the sense o f linear algebraic groups. The quotient G/ ( G, G ) is a torus. W e denote H = ( G, G ) and T = G/ ( G, G ), and o bta in a split exact sequence 1 → H ( A ) → G ( A ) → T ( A ) → 1 for any k -a lgebra A . Assuming that A is smo oth and essentially of finite type, we hav e an iso morphism T ( A ) ∼ = T ( A [ t ]). F rom the Ho chsc hild-Se r re sp ectral seq uence for the ab ov e group extensions we conclude that G ( k ) → G ( k [ t ]) induces an isomor - phism on homology if H ( k ) → H ( k [ t ]) induces an iso morphism o n homolog y . But H = ( G, G ) is a semisimple a lgebraic gr oup over k .  Prop osition 2.2. T o pr ove The or em 1.1, it su ffic es t o c onsider t he c ase wher e G is almost simple simply-c onne cte d over k . Pr o of. F o r a semisimple g roup G , there is an exa c t sequence of a lgebraic g roups 1 → Π → e G → G → 1 , ON HOMOLOGY OF LINEAR GR OUPS OVER k [ t ] 3 where Π is a finite central gro up scheme, and e G is a pro duct of simply-connected almost simple gro ups . If we assume tha t Theorem 1.1 holds for these simply- connected a lmost simple gro ups, it is also true for their pro duct, by a simple application of the Ho chsc hild-Serre sp ectra l sequence. Now assume that the order of Π is prime to the characteristic o f the field k , as in Theorem 1.1. F ro m the universal cov ering ab ov e we hav e an exact sequence 1 → Π( R ) → ˜ G ( R ) → G ( R ) → H 1 ´ et ( R, Π) → H 1 ´ et ( R, ˜ G ) for a n y k -a lgebra R . Then we have isomo rphisms Π( k ) ∼ = Π( k [ t ]) , and H 1 ´ et ( k , Π) ∼ = H 1 ´ et ( k [ t ] , Π) , by our as sumption on the characteristic of the bas e field. Now the firs t part of the exact sequence ab ov e yields a morphism of exact se- quences 1 / / Π( k ) / / ∼ =   ˜ G ( k ) / /   ˜ G ( k ) / Π( k ) / /   1 1 / / Π( k [ t ]) / / ˜ G ( k [ t ]) / / ˜ G ( k [ t ]) / Π( k [ t ]) / / 1 Since Π is in fact a belia n, one can consider fibre sequences B ˜ G ( R ) → B  ˜ G ( R ) / Π( R )  → K (Π( R ) , 2) for R = k and R = k [ t ]. Then the asso cia ted Ho chschild -Serre sp ectral seq uence implies that the mo rphism ˜ G ( k ) / Π( k ) → ˜ G ( k [ t ]) / Π( k [ t ]) induces an isomorphism on homology , since we ar gued b efor e that the morphism ˜ G ( k ) → ˜ G ( k [ t ]) induces an iso morphism o n homolo gy . Since H 1 ´ et ( k , ˜ G ) → H 1 ´ et ( k [ t ] , ˜ G ) is injective ( k is a re tract of k [ t ]), the ima g es of G ( k [ t ]) a nd G ( k ) in H 1 ´ et ( k , Π) ∼ = H 1 ´ et ( k [ t ] , Π) a r e equa l - we deno te these images by π 0 ( G ( k [ t ])) and π 0 ( G ( k )), re- sp ectively . Ther efore we get a morphism of ex tensions 1 / / ˜ G ( k [ t ]) / Π( k [ t ]) / /   G ( k [ t ]) / /   π 0 ( G ( k [ t ])) / / ∼ =   1 1 / / ˜ G ( k ) / Π( k ) / / G ( k ) / / π 0 ( G ( k )) / / 1 The outer vertical arrows are is omorphisms on homolog y , ther efore the compari- son theo rem for the Ho chsc hild-Ser re spe ctral s e q uences implies that we obta in an isomorphism o n the middle a rrow.  Henceforth, w e s hall only consider linear alg ebraic groups G , defined ov er k which are almost simple a nd simply- connected. 3. Bruha t-Tits buildings and the Sou l ´ e-Mar ga ux-theorem In this sec tio n, w e recall the bas ics of the theory of buildings whic h will be needed in the remaining sec tio ns. The main r eferences ar e [BT72] a nd [AB08]. Let k b e a field. Then we e quip the function field K = k ( t ) with the v aluation ω ∞ ( f /g ) = deg( g ) − deg( f ), with t − 1 as unifor mizer. W e deno te by O the co rre- sp onding discrete v aluation ring. Alternatively , one can w ork with K = k (( t − 1 )) and the cor resp onding v aluation ring k [[ t − 1 ]]. The underlying simplicial complex of the building will b e the sa me, only the apartment system will b e different. 4 MA TTHIAS WENDT Let G b e a reductive g roup ov e r k . Then we have tw o mor phisms of groups, the inclusion G ( O ) ֒ → G ( K ) and the reduction G ( O ) → G ( k ). 3.1. BN-Pairs and buil dings. W e will be concerned with affine buildings ass o - ciated to r eductive groups ov er discr etely v alued fields. W e recall the definition of buildings ba sed on the notion of BN-pair s. This theory is detaile d in [AB08], in particular Se c tio n 6 . Definition 3.1 . A p air of s ub gr oups B and N of a gr oup G is c al le d a BN-pair if B and N gener ate G , t he interse ction T := B ∩ N is normal in N , and the quotient W = N / T admits a set of gener ators S such that t he fol lowing two c onditio ns hold: (BN1) F or s ∈ S and w ∈ W we have sB w ⊆ B sw B ∪ B w B . (BN2) F or s ∈ S , we have sB s − 1  B . The gr oup W is c al le d the W eyl gr oup of the BN-p air. T he t u ple ( G, B , N , S ) is c al le d Tits system . W e now describ e the BN-pair o n G ( K ) whic h will be r elev ant for us. W e mos tly stick to the notation used in [Sou7 9]. Cho ose a maximal tor us T ⊆ G . This fixes t wo subgroups T ( k ) ⊆ G ( k ) and T ( K ) ⊆ G ( K ). Fix a choice of Borel subgro up B in G ( k ) co n taining T ( k ). F or the definition of the BN-pa ir, w e let B ⊆ G ( K ) be the pr eimage of B under the reduction G ( O ) → G ( k ). The gr oup N is defined as the no rmalizer o f T ( K ) in G ( K ). This is the usua l construction, ex plained in detail for the ca se S L n in [AB08, Section 6.9]. W e will not recall the pro of that this indeed yields a BN-pair here. W e reca ll one particular descr iption of the building ass o ciated to a BN-pair fro m [AB08, Section 6.2.6]. Given a Tits sy stem ( G, B , N , S ), a subgroup P ⊆ G is called p ar ab olic if it contains a conjugate of B . The subgroups of G which contain B are called st andar d p ar ab olic sub gr oups . These ar e ass o c iated to subsets of S . The building ∆( G, B ) for ( G, B , N , S ) is the simplicial complex asso ciated to the ordered set of parab olic subgr oups of G , ordered by r everse inclusion. The gr o up G a cts via conjuga tion. The fundamental apar tmen t is given b y Σ = { w P w − 1 | w ∈ W , P ≥ B } . The other apartmen ts are o f course obtained b y using conjugates of the group B abov e. Alterna tiv ely , the building can be describ ed as the simplicial co mplex asso ciated to the ordered set of cosets of the standar d par ab o lic subgro ups, with the g roup G acting via multiplication. 3.2. Soul´ e’s fundamental dom ain. W e contin ue to consider the B N-pair defined ab ov e . In the standa rd apartment Σ of ∆( G, B ), ther e is one vertex fixed by G ( O ). This vertex is denoted by φ . The fundamental cham b er containing the vertex φ is given by C = { P | P ≥ B } ⊆ Σ . The fundamen tal sector Q is the simplicial cone with vertex φ which is gener ated by C . The following theorem was prov ed in [Sou79] and subsequently g eneralized to isotropic s imply - connected a bs olutely almo st simple gr oups, cf. [Mar09]. Theorem 3.2. The set Q is a simpl icial fundamental do main for the action of G ( k [ t ]) on the Bruhat-Tits building ∆( G, B ) . In other wor ds, any simplex of ∆( G, B ) is e quivalent under the action of G ( k [ t ]) to a unique s implex of Q . ON HOMOLOGY OF LINEAR GR OUPS OVER k [ t ] 5 3.3. Stabili zers. W e are also in terested in the subgroups which stabilize simplices in the fundamental domain Q . W e hav e seen a bove that the simplices co rresp ond to standard par a bo lic subgr oups. It turns o ut tha t the stabilizer of the s implex corres p onding to G ≥ P ≥ B is exactly P , cf. [AB08, Theorem 6.43 ]. In particula r, for the gro up G ( k [ t ]), we find that the stabilizer of a simplex σ P corres p onding to a pa r ab olic subgro up P of G ( K ) is the following intersection Stab( σ ( P ) ) = G ( k [ t ]) ∩ P . This implies a conc r ete descr iption of the stabilizers, cf. [Sou79, Paragraph 1.1] resp. [Mar09, Prop osition 2.5]. Prop osition 3.3. L et x ∈ Q \ { φ } . W e denote by Stab( x ) the stabilizer of x in G ( k [ t ]) . (i) Ther e is an extension of gr oups 1 → Stab( x ) ∩ U x ( K ) → Stab( x ) → L x ( k ) → 1 . The gr oup L x ( k ) is a r e ductive sub gr oup of G ( k ) , in fact it is a L evi sub gro up of a maximal p ar ab olic sub gr oup of G ( k ) for the spheric al BN-p air. The gr oup Stab( x ) ∩ U x ( K ) is a split unip otent sub gr oup of U x ( k [ t ]) . (ii) The st abilizer of a simplex σ is the interse ction of the st abilizers of t he vertic es x of σ . (iii) The stabilizers c an b e describ e d using the valuation of the r o ot system, cf. [Sou79, Section 1.1] . In p articular, in the notation of Soul´ e , we have Γ x = L x ( k ) · U x ( k [ t ]) , L x ( k ) = T ( k ) · h x α ( k ) | α ( x ) = 0 i , U x ( k [ t ]) = h x α ( u ) , u ∈ k [ t ] , d ◦ ( u ) ≤ α ( x ) , α ( x ) > 0 i . Withouth explaining all the notation in detail, this me ans that an element of Z x ( k ) = L x ( k ) is a pr o duct of an element of the torus and c ertain r o ot elements, wher e t he r o ots only dep end on the vertex x . An element in U x ( k [ t ]) is a pr o duct of c ertain r o ot element s, the de gr e e of t he p olynomials and the r o ots only dep end on the vertex x . 4. Homol ogy of the st abilizers and Kn u dson’s theorem In this section, we describ e the homolog y o f the stabilizers of simplices in So ul ´ e’s domain. In [Knu97], Knudson showed in the c ase S L n that the ho mo logy o f the stabilizers is determined by the homology of a Levi s ubg roup. W e pr ovide b elow a generaliza tion of this re s ult. The results w ork in gener al for rings with man y units. F rom Pr o po sition 3.3, we know that for a simplex σ in the fundamental domain Q , the stabilize r Γ σ = Stab( σ ) of σ in G ( k [ t ]) sits in an extension 1 → U σ → Γ σ → L σ → 1 , where U σ is an abs tract gro up contained in a unip otent subgro up of G ( k [ t ]) and L σ is the group of k -p oints of a reductive subgr o up o f G . The first thing we will show in this section is that the induced mor phism H • (Γ σ , Z ) → H • ( L σ , Z ) is an isomorphism. This is done via the Ho chschild-Serre sp e c tral se q uence E 2 p,q = H p ( L σ , H q ( U σ )) ⇒ H p + q (Γ σ ) asso ciated to the ab ov e g roup extension. T o show the r esult, it suffices to show that H p ( L σ , H q ( U σ )) = 0 for q > 0. The basic idea for showing this latter as sertion is to use the a c tion o f k × on the group U σ , where k × is em bedded in L σ as the k -p oints of a suita ble subtorus. The group k × acts via multiplication on the v arious ab elian sub quotients co nstituting the unip otent gr oup U σ , a nd an argument as in [Knu01, Theorem 2.2 .2 ] shows that 6 MA TTHIAS WENDT this homology is trivia l. This argument is detailed after some introductor y remarks in Theorem 4.6. 4.1. A result of Suslin. A ring A is an S ( n )-ring if there are a 1 , . . . , a n ∈ A × such that the sum of each no nempt y s ubfamily is a unit. If A is an S ( n )-ring for all n , then A is said to have many units . As explaine d in [Knu01, Sectio n 2.2.1 ], the rig ht wa y to pr ov e tha t H p ( GL n ( A ) , H q ( M n,m ( A ) , Z )) = 0 for q > 0 if A is a Q -a lgebra is the fo llowing: we notice that M n,m ( A ) is an abe lian group, and therefor e H q ( M n,m ( A )) = V q M n,m ( A ). There exists a central element a ∈ GL n ( A ) which acts o n M n,m ( A ) by multiplication with a , and there fore by m ultiplication with a q on H q ( M n,m ( A )). This a ction is triv ial, b ecause a is in the center, a nd therefore H q ( M n,m ( A )) is annihilated by a q − 1. But it is a Q -vector space and there fo re it is trivial for q > 0. The following result due to Nesterenko a nd Suslin [NS90] is a generaliza tion of this center-kills-argument to ring s with man y units in ar bitr ary c haracteristics. F or more information on the pr o of of this result, cf. [Knu01, Section 2.2 .1]. Prop osition 4.1. L et A b e a ring with many units, and let F b e a prime field. Then for al l i ≥ 0 and j > 0 , we have H i ( A × , H j ( A s , F )) = 0 , wher e A × acts diagonally on A s . The same c o nclusion a lso obtains for actions of A × via no n-zero p ow er s of units, cf. [Hut90, Lemma 9]. 4.2. Example : orthogonal g roups. W e explain the pr o cedure using the sp ecial case of o rthogonal g roups from [V og7 9]. F or the groups O n,n ov er a field k of characteristic 6 = 2, ther e are maximal parab olic subgro ups P I which hav e a non- ab elian unip otent radical. They have the following ge ne r al fo r m, cf. [V o g79, p. 21]:   A ∗ ∗ 0 B ∗ 0 0 t A − 1   , where A ∈ GL p ( k ), B ∈ O n − p,n − p ( k ) a nd ther e are so me additional co nditions on the ∗ -terms ensur ing that the whole matrix is in O n,n ( k ). It is pr oved on p.34 of that pap er that the unip otent subgr oup N of P I sits in a n exact sequence 1 → [ N , N ] → N → N / [ N , N ] → 1 with the o uter ter ms [ N , N ] and N / [ N , N ] ab elian gro ups. It is also prov ed that the to rus diag( a, . . . , a | {z } p , 1 , . . . , 1 | {z } 2 n − 2 p , a − 1 , . . . , a − 1 | {z } p ) acts v ia multiplication with a on N / [ N , N ] and multiplication with a 2 on [ N , N ]. W e apply the Ho chsc hild-Serre sp ectral sequence for the extension 1 → [ N , N ] → P I → P I / [ N , N ] → 1 . This has the fo llowing form: H p ( P I / [ N , N ] , H q ([ N , N ])) ⇒ H p + q ( P I ) . T o prov e that H p ( P I / [ N , N ] , H q ([ N , N ])) = 0 for q > 0, we use another Ho chsc hild- Serre sp ectral sequence for the torus action: H p ( P I / [ N , N ] /k × , H j ( k × , H q ([ N , N ])) ⇒ H p + j ( P I / [ N , N ] , H q ([ N , N ])) . ON HOMOLOGY OF LINEAR GR OUPS OVER k [ t ] 7 Since the action of k × on [ N , N ] is via multiplication by squares , the result of Suslin, cf. Prop ositio n 4.1, implies H p ( P I / [ N , N ] , H q ([ N , N ])) = 0 for q > 0. The morphism P I → P I / [ N , N ] th us induces an isomor phism on homo logy . A similar argument applied to the extens io n 1 → N / [ N , N ] → P I / [ N , N ] → P I / N → 1 implies that the morphism P I / [ N , N ] → P I / N also induces an isomo rphism o n homology . This a moun ts to a pro of of [V og 7 9, P rop osition 2 .2 ] for infinite fields of char- acteristic p 6 = 2. W e obtain the following streng thening of V ogtmann’s stability result, ma king explicit a re mark in [Knu01, Section 2.4.1 ]. Corollary 4. 2. L et k b e an infinite field of char acteristic 6 = 2 . Then the induc e d morphism H i ( O n,n ( k ) , Z ) → H i ( O n +1 ,n +1 ( k ) , Z ) is su rje ctive for n ≥ 3 i + 1 and an isomorphi sm for n ≥ 3 i + 3 . Remark 4 . 3. (i) The ab ove st abilization r esult is the one obtaine d via V o gt- mann ’s ar gument in [V og 79] using the impr ove d c omputation of the homolo gy of st abilizers. Bett er stabilization r esults for ortho gonal gr oups ar e availabl e in the work of Essert, cf. [Ess09] . In Essert’s work, de aling with t he homo- lo gic al c ontribution fr om t he u nip otent r adic al as ab ove is not ne c essary - he uses opp osition c omplexes, wher e stabilizers ar e L evi sub gr oups inste ad of the ful l p ar ab olic su b gr oups. (ii) The ab ove example for ortho gonal gr oups is an inst anc e of a mor e gener al r esult which c an b e found e.g. in [ABS9 0] . L et G b e a r e ductive gr oup over k , let Φ b e the asso ciate d r o ot system and assume that char k is not e qual to 2 for Φ doubly lac e d r esp. not e qual to 2 or 3 for Φ triply lac e d. L et P b e a p ar ab olic s u b gr oup asso ciate d to a subset I ⊆ Φ of simple r o ots, and let U b e the u nip otent r adic al of P . Then the length of the desc ending c ent r al series of U e quals P α ∈ I m ( α ) wher e α is the multiplicity of α in the highest r o ot e α of Φ . In t he ab ove example of ortho gonal gr oups, t he ro ot system is of t yp e D n . Numb ering the simple r o ots α 1 , . . . , α n − 1 , α n such that α 1 , α n − 1 and α n c or- r esp ond t o the end-p oints of the Dynkin diagr am, the longest r o ot is e α = α 1 + 2 α 2 + · · · + 2 α n − 2 + α n − 1 + α n . The p ar ab olic sub gr oups discusse d in t he ab ove example ar e the ones c orr e- sp onding to r o ots α 2 , . . . , α n − 2 . 4.3. Hom ology o f the stabilizers. W e now want to co mpute the ho mology of the s tabilizers. W e for m ulate the pro of with the k -p oints o f the stabilizer s, where k is an infinite field. The same arguments show the r esult mo re g enerally for an int egral doma in R with ma ny units ha ving quotient field k . The goal is to co mpute the ho mology of the stabilizer Γ σ ( k [ t ]). The Levi subgroup L σ is defined as L σ = Z G ( S σ ), i.e. as the centralizer of a split torus S σ in G ass o c ia ted to the simplex σ . W e note that it follows from this definition that there is a normal central tor us in L σ . Now cons ider a subtor us G m → L σ . If the corres p onding a bstract group k × acts trivially on the k -p oints of a unip o ten t s ubgroup U ⊆ G ( k [ t ]), then U ⊆ Z G ( G m ) and hence U is contained in L σ . Therefore, fo r a n y unip otent subgro up U ⊆ Γ σ which is not contained in L σ , there has to exist a torus G m ⊆ L σ such that the corres p onding group o f k - po ints k × acts no n trivially on the k -p oint s o f U . Note that the unip otent radica ls U σ of para bo lic s ubgroups of G a sso ciated to the s implex σ are ac tually split, i.e. there is a filtration U σ = U 1 ⊇ U 2 ⊇ · · · ⊇ U n = { 1 } 8 MA TTHIAS WENDT with e a ch U n /U n +1 being isomorphic to G a . Since the automorphism gro up of G a is G m , the mu ltiplicative group G m can o nly act via ( a ∈ k × , u ∈ k ) 7→ a n u for s ome n . W e have thus e stablished the following: Lemma 4.4 . L et U σ b e the unip otent r adi c al of the stabilizer Γ σ , and let U σ = U 1 ⊇ U 2 ⊇ · · · ⊇ U n = { 1 } b e a fi ltr ation s uch t hat U i /U i +1 ∼ = G a . F or e ach i , ther e exists a c entr al emb e dding k × → L σ and a nu mb er n i such that a ∈ k × acts on U i /U i +1 via multiplic ation with a n i . Note that above we are only talking ab out alge br aic g r oups, i.e. ab o ut the unipo ten t radica l o f parab olic subgroups of G ( k ( t )). Howev e r , since the k [ t ]-p oints of the tor us a re k × , the action pre s erves the degree filtra tion of the k [ t ]-p oints of unipo ten t radica ls. In par ticula r, the action descr ibed a bove restr icts to an a ction of k × on the unip otent part U σ of the stabilizer subgroup Γ σ , for an y simplex σ ∈ Q . Example 4. 5. (i) The simplest example of this situ ation is the emb e dding R × → S L n + m : a 7→ diag( a m , . . . , a m , | {z } n a − n , . . . , a − n | {z } m ) . The c entr alizer of t his torus is the L evi su b gr ou p of a maximal p ar ab olic sub- gr oup which is the int erse ction of t he fol lowing sub gr oup with S L n + m :  GL n 0 0 GL m  . The c orr esp onding p ar ab olic s ub gr oup has the form  GL n M 0 GL m  ∩ S L n + m and the torus acts on M via multiplic ation with a m + n , cf. [Hut90] . (ii) Another example of such a situation is the one discusse d in t he pr o of of Cor ol lary 4.2, cf. also [V og79, p. 34] . In these c ases, the unip otent r adi - c al of a maximal p ar ab olic of a split ortho gonal or sympl e ctic gr oup is not ab elian, and the torus acts via differ ent p owers on the steps of the c entr al series.  The ab ov e actio ns now a llow to co mpute the E 2 -term of the Ho chsch ild-Serre sp ectral seq uence. This is done b y using the co mpos ition series of U I , which induces a se q uence Γ I → Γ I / G a = Γ I /U n → Γ I /U 2 → · · · → Γ I /U I = L I . W e will show b elow that each s tep induces isomo rphisms on homolo gy . The argu- men t is a gener alization of the pro o f of [Knu97, Coro llary 3.2]. The following theorem now describ es the homology o f the s tabilizers of the action of G ( k ) on the Bruha t-Tits building. Theorem 4 . 6. L et R b e an inte gr al domain with many units and denote by k = Q ( R ) its field of fr actions. The gr oup G ( R [ t ]) acts on the Bruhat-Tits building asso ciate d to the gro up G ( k ( t )) , and we c onsider the stabilizer gr oup Γ σ of a simplex σ ∈ Q . Then the morphism H • (Γ σ , Z ) → H • ( L σ , Z ) induc e d fr om the pr oje ction in Pr op osition 3.3 is an isomorphism. ON HOMOLOGY OF LINEAR GR OUPS OVER k [ t ] 9 Pr o of. Consider the comp ositio n series of U σ : U σ = U 1 ⊇ U 2 ⊇ · · · ⊇ U n = { 1 } . More precis ely , the comp ositio n series o f the unip otent gr oup as a n algebr aic group induces a s imilar filtration o f the unipo ten t pa rt of the stabilize r, which is defined inside the unip otent ra dical by degree b ounds as in P rop osition 3.3. This induces a se q uence of gr oup homomor phis ms Γ σ → Γ σ /U n → Γ σ /U 2 → · · · → Γ σ /U σ = L σ . Each step in this s e quence is a quotient by a subgr oup o f G a ( R [ t ]) in Γ σ /U i . It therefore suffices to sho w that each such mo rphism induces an isomorphism on homology . This is do ne via the Hochsc hild-Serre sp ectral seque nc e , whic h then lo oks like H p (Γ σ /U i +1 , H q ( U i /U i +1 , Z )) ⇒ H p (Γ σ /U i , Z ) . Thu s it suffices to show for any prime field F , we hav e H p (Γ σ /U i +1 , H q ( U i /U i +1 , F )) = 0 for q > 0. But by Lemma 4 .4, ther e is a central embedding R × → Γ σ /U i +1 such that a ∈ R × acts o n U i /U i +1 via multiplication by some no n- zero p ow er of a . W e hav e an a sso ciated Ho chschild-Serre sp ectra l sequence E 2 j,l = H j ((Γ σ /U i +1 ) /R × , H l ( R × , H q ( U i /U i +1 , F ))) ⇒ ⇒ H j + l (Γ σ /U i +1 , H q ( U i /U i +1 , F )) . F rom Pro po sition 4.1, we obtain that H l ( R × , H q ( U i /U i +1 , F )) = 0 for q > 0, whic h finishes the pr o o f.  4.4. The theorem o f Kn udson. W e will now prov e homoto p y inv ar iance in the one-v ar iable case. The following is a generaliza tion of [K n u01, Coro llary 4.6.3]. Theorem 4.7. L et k b e an infinite field and let G b e a c onne cte d r e ductive gr oup over k . Then the inclusion k ֒ → k [ t ] induc es an isomorphism H • ( G ( k ) , Z ) ∼ = − → H • ( G ( k [ t ]) , Z ) , if the or der of t he fundamental gr oup of G is invertible in k . Pr o of. By P rop osition 2.2 , we can assume that G is simply-co nnected abso lutely almost simple ov er k . The n we can use the action of the gr o up G ( k [ t ]) on the build- ing a sso ciated to G ( k ( t )). By Theorem 3.2, the sub complex Q is a fundamental domain for this a ction. There is a n ass o c iated sp ectral sequence E 1 p,q = M dim σ = p,σ ∈Q H q (Γ σ , Z ) ⇒ H p + q ( G ( k [ t ]) , Z ) . In the a bove, Γ σ is the stabilizer of the simplex σ . This is the sp ectral sequence computing the G ( k [ t ])-equiv ariant homology o f the building, cf. [K n u01, p. 162 ]. F rom Theorem 4.6, w e know the homo logy of the stabilizers, in particular, that it only dep ends o n the r eductive part. In the notation of [Knu97], there is a filtr ation of the fundament al domain Q via subs e ts E ( k ) I for a n y k -element subset I of ro ots of G . These subsets are simplicial sub cones of Q which consist of all simplices of Q such that the co nstant part of the stabilizer is the standard parab olic subgr o up of G determined by the s ubset I . This yields a filtration Q ( k ) = [ I E ( k ) I . 10 MA TTHIAS WENDT Now for any tw o simplices σ , τ in E ( k ) I \ [ J ⊂ I E ( k − 1) J , the stabilizers Γ σ and Γ τ hav e the same reductive part L σ = L τ , and therefo r e they also ha ve the sa me homolog y , b y Theo rem 4 .6. The co efficient system σ 7→ H q (Γ σ ) is then lo ca lly constant in the s ense of [Knu01, Prop ositio n A.2.7 ], and w e obtain an iso morphism H • ( φ, H q ) → H • ( Q , H q ) . This shows that the argument in the pro o f of [Knu97, Theor em 3.4] does not depend on S L n . Therefore, [Knu01, P rop osition A.2.7] implies that the E 2 -term o f the ab ov e sp ectral sequence lo oks as follows: E 2 p,q =  H q ( G ( k ) , Z ) p = 0 0 p > 0 The sp ectral sequence degenerates and the r esult is proved.  Remark 4. 8 . The or em 4.6 has b e en use d in [W en12] to establish homotopy invari- anc e for the homolo gy of Steinb er g gr oups of r ank two gr oups. Ap art fr om t his, it se ems that the adde d gener ality of rings with many un its in The or em 4.6 c an not b e widely applie d. Gener alizatio ns of The or em 4.7 b eyond the c ase k [ t ] se em to b e gener al ly wr ong. The failur e of homotopy invarianc e for H 1 of S L 2 fol lows dir e ctly fr om [KM97] . The failur e of homotopy invarianc e for H 2 of r ank two gr oups has b e en establishe d in [W en1 2] . In these c ases, one se es that Q fails quite b ad ly to b e a fundamental domain for the action of G ( R [ t ]) if R is not a field – in c ase S L 2 , the su b c omplex S L 2 ( R [ t ]) · Q is not c onne cte d and in c ase S L 3 , the sub c omplex S L 3 ( R [ t ]) · Q is not simply-c onne cte d. Concerning the s ub complex G ( R [ t ]) · Q for R an int egral domain, we hav e the following: Prop osition 4. 9. (i) The c omplex E ( R [ t ]) · Q is c onne cte d. (ii) The c omplex G ( R [ t ]) · Q is c onne cte d if G ( R [ t ]) = E ( R [ t ]) · G ( R ) . This in p articular holds for isotr opic r e ductive gr oups G of r ank ≥ 2 and R essent ial ly smo oth over a field. (iii) The c omplex E ( R [ t ]) · Q is simply-c onne cte d if K G 2 ( R [ t ]) ∼ = K G 2 ( R ) , in p artic- ular for G = S L n , n ≥ 5 and R = k [ t 1 , . . . , t m ] . Pr o of. Every elementary matrix for a p ositive ro o t is contained in some stabilizer , and the stabilizer of φ contains the W e y l group. 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