Absence of torsion for NK_1(R) over associative rings

Absence of torsion for NK 1 ( R ) o v er asso ciativ e rings Rab ey a Basu Dep artmen t of Mathematic al Scienc es, Indian Institute of Scienc e Educ at ion and R ese ar ch, Kolkata, West Bengal, India rb asu@iiserkol.ac.i n 2000 Mathematics Su bje ct Classific ation:13H99, 15A24, 16R50, 19B14 Key wor ds: line ar gr oup, K 1 , NK 1 , SK 1 , torsion, Witt ve ctors. Abstract When R is a commutativ e r ing with iden tity , and if k ∈ N , with k R = R , then it was shown in [11] that SK 1 ( R [ X ]) has no k-torsion. W e repr ov e this result for a ny asso ciative ring R with iden tity in which k R = R . 1 In tro duc tion Let R be an as so ciative ring with identit y element 1. Let K 1 ( R ) deno te the White- head group. In ca s e R is commutativ e, let SK 1 ( R ) b e the kernel of the determinant map from K 1 ( R ) to the group of units of R . Let W( R ) b e the ring of big Witt vectors. W e denote NK 1 ( R ) = ker (K 1 ( R [ X ]) → K 1 ( R )); X = 0 . In [10], J. Stienstra, using ideas o f S. Blo ch in [2], show ed that NK 1 ( R ) is a W( R )-mo dule. Consequently , as noted b y C. W eibe l in ([1 1], § 3), if k is a unit in R , then SK 1 ( R [ X ]) ha s no k -to rsion, when R is a commutativ e lo cal r ing. F or a comm utative ring with identit y NK 1 ( R ) coincides with SK 1 ( R [ X ]) if we take R to b e lo cal. In this note we generalize W eib el’s observ ation for NK 1 ( R ), where R is a commutativ e lo ca l ring. W e prov e for an asso ciative ring R with identit y , NK 1 ( R ) has no k -tors ion if k is a unit in R . In particular , this shows W eib el’s result is a spe c ial cas e. The metho d of pro o f ma y b e considered as a simplified version of J. Stienstra ’s approa ch via big Witt vectors. This will help the r eader to apprecia te w hy big Witt vectors co me int o the pictur e in the functorial approach of S. B lo ch, et al . Theorem 1.1 L et R b e an asso ciative ring with identity. If k is a unit in R , then NK 1 ( R ) has no k -torsion. Corollary 1.2 L et R b e a c ommutative lo c al ring with identity. If k is a un it in R , then SK 1 ( R [ X ]) has no k - torsion. As a consequence of Theorem 1 we prov e 1 Theorem 1.3 L et R = R 0 ⊕ R 1 ⊕ · · · b e a gr ade d c ommutative ring with identity. L et k b e a u n it in R 0 . L et N = N 0 + N 1 + · · · + N r ∈ M r ( R ) b e a nilp otent matrix. If [( I + N )] k = [ I ] in SK 1 ( R ) , then [ I + N ] = [ I + N 0 ] . In p articular, if R 0 is a r e duc e d lo c al ring, then SK 1 ( R ) has no k -torsion. 2 Prologue Let R b e an asso cia tive ring with identit y . GL n ( R ) denotes the g roup of inv ertible matrices, SL n ( R ) its subgroup of ma trices of determinant 1 (when R is a co mm utative ring), E n ( R ) the subgr oup of e lementary ma trices, i.e. generated by { E ij ( λ ) : λ ∈ R, i 6 = j } , where E ij ( λ ) = I + λe ij and e ij is the ma trix with 1 on the ij -th position and 0’s elsewhere. F or α ∈ M r ( R ), β ∈ M s ( R ) w e have α ⊥ β ∈ M r + s ( R ), where α ⊥ β =  α 0 0 β  . There is an infinite counterpart: iden tifying each matrix α ∈ GL n ( R ) with the large matrix ( α ⊥ 1) gives an embedding o f GL n ( R ) in to GL n +1 ( R ). L e t GL( R ) = ∪ ∞ n =1 GL n ( R ), SL( R ) = ∪ ∞ n =1 SL n ( R ), and E( R ) = ∪ ∞ n =1 E n ( R ) be the corr esp onding infinite linea r gr oups. The well known Whitehead’s L emma asserts that if α ∈ GL n ( R ) then we hav e ( α ⊥ α − 1 ) ∈ E 2 n ( R ). Thus we hav e [GL( R ) , GL( R )] = [E( R ) , E( R )] = E( R ) and hence E( R ) is a normal subgroup of GL( R ). The quo tient GL( R ) / E ( R ) is called the W hitehead g roup of the r ing R and is deno ted by K 1 ( R ). F or α ∈ GL n ( R ) let [ α ] denote its equiv alence class in K 1 ( R ). Also, as a consequence of Whitehead’s lemma one sees that if α, β ∈ GL n ( R ) then [ α, β ] ∈ E 2 n ( R ); whence K 1 ( R ) is an ab elian g roup. F or details cf. [1 ]. In case R is commutativ e the determinant map from GL n ( R ) to R ∗ induces a map, det : K 1 ( R ) → R ∗ given by αE ( R ) 7→ det α . The kernel of the map is denoted by SK 1 ( R ) and equals to SL( R ) / E( R ). W e wr ite NK 1 ( R ) for ker(K 1 ( R [ X ]) → K 1 ( R )); X = 0, i.e. the s ubgroup con- sisting of elements [ α ( X )] ∈ K 1 ( R [ X ]) such that [ α (0)] = [ I ]. Note that if R is a commutativ e lo cal ring then SK 1 ( R [ X ]) coincides with NK 1 ( R ); indeed, if R is a lo cal r ing then SL n ( R ) = E n ( R ) for all n > 0. Therefore, we may r eplace α ( X ) by α ( X ) α (0) − 1 and assume that [ α (0 )] = [ I ]. F or a commutativ e r ing R the group W( R ) of big Witt v ectors is defined by: W( R ) = (1 + X R [[ X ]]) × . F or P ( X ) ∈ (1 + X R [[ X ]]), let ω ( P ) denote the cor resp onding element of W( R ). The group structur e in W( R ) is given by: ω ( P ) + ω ( Q ) = ω ( P .Q ) . An y P ( X ) ∈ (1 + X R [[ X ]]) can b e written uniquely as a pro duct: P ( X ) = Π n ≥ 1 (1 − a n X n ) − 1 , a n ∈ R. 2 The elements ( a 1 , a 2 , . . . . . . ) ar e called the Witt co-o rdinates of ω ( P ). Also, there exists a unique s tructure o f commutativ e r ing o n W( R ) such tha t ω ((1 − aX m ) − 1 ) .ω ((1 − bX n ) − 1 ) = ω ((1 − a n/r b m/r X mn/r ) − r ) , where r = g.c.d( m, n ). The identit y element in W( R ) is presented by the p ow er series (1 − X ) − 1 . F or details cf. ([2], Pro p. I.I), [7]. F or the W( R )-mo dule structure o f NK 1 ( R ) see [1 2] (for previous articles cf. [2 ], [3], [10], [11]). 3 Higman Linearization Two matrices α ∈ M r ( R ) and β ∈ M s ( R ) are said to be stably equiv alent if there exists ε 1 , ε 2 ∈ E t ( R ) (for some t ≥ max { r, s } ) s uch that ε 1 ( α ⊥ I t − r ) ε 2 = ( β ⊥ I t − s ). Lemma 3.1 (Higman Linearization Pro cess) L et α ( X ) b e a matrix over R [ X ] . Then α ( X ) is stably e quivalent to a line ar matrix in M s ( R [ X ]) for some s . Pro of. W e may assume that n ≥ 2. Let α ( X ) = a 0 + a 1 X + a 2 X 2 + · · · + a n X n ∈ M r ( R ) , r > 1 . Then α ( X ) is stably equiv alent to a matrix of degr ee n − 1 over R [ X ] in the following manner:  I r − a n X 0 I r   a 0 + · · · + a n X n 0 0 I r   I r 0 X n − 1 I r I r  =  a 0 + · · · + a n − 1 X n − 1 − a n X X n − 1 I r I r  = α 1 has deg ree ( n − 1). Hence α is stably equiv alent to α 1 . Rep eating the above pro cess ( n − 2 ) times we get the result. ✷ Corollary 3.2 L et R b e an asso ciative ring with identity. L et α ( X ) ∈ GL r ( R [ X ]) with α (0) = I n . Then in K 1 ( R [ X ]) we have [ α ( X )] = [ I s + N X ] for some s > 0 and some matrix N ∈ M s (R) . Pro of. By Lemma 3.1 ther e exists ε 1 , ε 2 ∈ E t ( R [ X ]) (for some t > r ) such that ( α ( X ) ⊥ I t − r ) = ε 1 (( I s + N X ) ⊥ I t − s ) ε 2 for some s > 0 a nd N ∈ M s (R). Now as E( R ) is a norma l subgroup of GL( R ), for some integer u there exists ε ′ 1 ∈ E t + u ( R [ X ]) such that ( ε 1 ⊥ I u )(( I s + N X ) ⊥ I t − s + u ) = (( I s + N X ) ⊥ I t − s + u ) ε ′ 1 . Hence in K 1 ( R [ X ]) we hav e [ α ( X )] = [(( I s + N X ) ⊥ I t − s + u ) ε ′ 1 ( ε 2 ⊥ I u )] = [ I s + N X ] . ✷ 3 4 Main Theorem Let R t denote the ring R [ X ] / ( X t +1 ). Lemma 4.1 L et R b e a ring and P ( X ) ∈ R [ X ] b e any p olynomial. Then the fol lowing identity holds in the ring R t : (1 + X r P ( X )) = (1 + X r P (0))(1 + X r +1 Q ( X )) , wher e r > 0 and Q ( X ) ∈ R [ X ] , with deg( Q ( X )) < t − r . Pro of. Let us write P ( X ) = a 0 + a 1 X + · · · + a t X t . Then we can wr ite P ( X ) = P (0) + X P ′ ( X ) for some P ′ ( X ) ∈ R [ X ]. Now, in R t (1 + X r P ( X ))(1 + X r P (0)) − 1 = (1 + X r P (0) + X r +1 P ′ ( X ))( 1 + X r P (0)) − 1 = 1 + X r +1 P ′ ( X )(1 − X r P (0) + X 2 r ( P (0)) 2 − · · · ) = 1 + X r +1 Q ( X ) where Q ( X ) ∈ R [ X ] with deg( Q ( X )) < t − r . Hence the lemma follows. ✷ Remark. Itera ting the a b ov e pro cess we can wr ite for any p o lynomial P ( X ) ∈ R [ X ], (1 + X P ( X )) = Π t i =1 (1 + a i X i ) in R t , for some a i ∈ R . By a scending induction it will follow that the a i ’s are uniquely determined. In fact, if R is commutativ e then a i ’s are the i -th component of the gho st vector corr esp onding to the big Witt vector of (1 + X P ( X )) ∈ W( R ) = (1 + X R [[ X ]]) × . F or details see ([2], § I). Lemma 4.2 L et R b e a ring with 1 k ∈ R and P ( X ) ∈ R [ X ] . Assum e P (0) lies in the c enter of R . Then (1 + X r P ( X )) k r = 1 ⇒ (1 + X r P ( X )) = (1 + X r +1 Q ( X )) in the ring R t for some r > 0 and Q ( X ) ∈ R [ X ] with deg( Q ( X )) < t − r . Pro of. By Le mma 4.1 (1 + X r P ( X )) = (1 + X r P (0))(1 + X r +1 P 1 ( X )) , (1) for so me P 1 ( X ) ∈ R [ X ] with deg ( P 1 ( X )) < t − r . Therefore, in R t (1 + X r P ( X )) k r = 1 ⇒ (1 + X r P (0)) k r = (1 + X r +1 P 1 ( X )) − k r . As 1 k ∈ R , w e have (1 + k r X r P (0) + X r +1 P 2 ( X )) = (1 + X r +1 P 1 ( X )) − k r . This implies (1 + k r X r P (0)) = (1 + X r +1 P 1 ( X )) − k r (1 + (1 + k r X r P (0)) − 1 X r +1 P 2 ( X )) − 1 = (1 + X r +1 P 3 ( X )) for some P 2 ( X ) , P 3 ( X ) ∈ R [ X ] with deg P 2 ( X ) , P 3 ( X ) < t − r . Now, applying homomorphism X 7→ 1 k X we get (1 + X r P (0)) = (1 + X r +1 P 4 ( X )) 4 for so me P 4 ( X ) ∈ R [ X ] with deg ( P 4 ( X )) < t − r . Substituting this in (1) we get (1 + X r P ( X )) = (1 + X r +1 Q ( X )) for so me Q ( X ) ∈ R [ X ] with deg( Q ( X )) < t − r . ✷ Pro of of Theorem 1.1. Let α ( X ) ∈ GL n ( R [ X ]) with [ α (0)] = [ I ] b e a k -tors ion. By Corollar y 3.2 in K 1 ( R [ X ]), [ α ( X )] = [( I s + N X )] for some s > 0 and N ∈ M s ( R [ X ]). Since ( I s + N X ) is in vertible, N is nilp o tent . Let N t +1 = 0. Since [( I s + N X )] k = [ I ] in K 1 ( R [ X ]), it follows that [ I s + k N X + N 2 X 2 P 1 ( N X )] = [ I ] for so me P 1 ( X ) ∈ R [ X ]. Hence, a s b efor e, as 1 k ∈ R , [( I s + k N X ) − 1 ] = [( I s + ( I s + k N X ) − 1 N 2 X 2 P 1 ( N X )] = [ I s + ( I s − k N X + N 2 X 2 P 2 ( N X )) N 2 X 2 P 1 ( N X )] = [ I s + N 2 X 2 P ( N X )] for so me P ( X ) ∈ R [ X ]. Since [( I s + N 2 X 2 P ( N X ))] k = [ I ], arguing a s in the pro of of Lemma 4.2 we get in K 1 ( R [ X ]) [ I s + N 2 X 2 P ( N X )] = [ I s + N 3 X 3 Q ( N X )] for so me Q ( X ) ∈ R [ X ]. Now by rep eating the ab ove mentioned ar gument we ge t [ I s + N 2 X 2 P ( N X )] = [ I ] . Finally , applying homo mo rphism X 7→ 1 k X we get the desired result. ✷ Now we prove Theore m 1.3 a s a consequence o f Swan-W eib el homotopy trick. F o r details see ([6], Pro of of Prop. 2.22). First w e recall the Lo cal-Global P rinciple fo r a graded ring . Graded Lo cal-Global Principle: Let R = R 0 ⊕ R 1 ⊕ · · · b e a graded commutativ e ring with k a unit in R 0 and α ( X ) ∈ GL n ( R [ X ]) with α (0) = I n . If α m ( X ) ∈ E n ( R m [ X ]) for all m ∈ Max ( R 0 ), then α ( X ) ∈ E n ( R [ X ]). This is derived from the usual Lo ca l- Global P rinciple by using the fo llowing ho - motopy tr ick due to Swan and W eib el. Pro of of Theorem 1. 3. C o nsider the r ing homomorphism θ : R → R [ X ], g iven by ( a 0 + a 1 + · · · ) 7→ a 0 + a 1 X + · · · . Then [( I + N )] k = [ I ] ⇒ θ ([( I + N )] k ) = [( θ ( I + N ))] k = [ I ] ⇒ [( I + N 0 + N 1 X + · · · + N r X r )] k = [ I ] Let m be a maximal ideal in R 0 . By Theorem 1 [( I + N 0 + N 1 X + · · · + N r X r )] = [ I ] in SK 1 ( R 0 ) m . Hence by the Gra ded Lo ca l-Global Principle [( I + N )] = [( I + N 0 )] in SK 1 ( R ). In particular, if R 0 is a reduced lo ca l r ing then units of R 0 [ X ] = units of R 0 . Since N 0 is nilpotent, det( I + N 0 X )= co nstant = 1. Hence I + N 0 is in SL r ( R 0 ) = E r ( R 0 ). This completes the pro of. ✷ 5 5 Examples W e show b y an example that the the condition 1 k ∈ R is necessa r y in Theorem 1.3, whence in Theorem 1 .1. Example. Let R b e a commutativ e ring with identit y and α a 2 × 2 co mpletion of (1 − X Y , X 2 ) over R [ X 2 , X Y , Y 2 ]. In ([5], § 8, Example 8.2 ) it is shown that α ∈ SK 1 ( R [ X 2 , X Y , Y 2 ]) \ SK 1 ( R ). Now ta ke R = Z 2 . Then the square o f the Mennic ke symbol of the vector (1 − X Y , X 2 ) is clear ly trivial, i.e. α 2 ∈ E 3 ( Z 2 [ X 2 , X Y , Y 2 ]). (See [1] for definition of Mennicke sym b ol.) Remark. Let P b e a finitely g enerated pro jective R - mo dule. A theorem of M.R. Gab el in [4] ass erts that mP = P ⊕ · · · ⊕ P (m times) is free, for some m . (A result of T.Y. Lam in [8] sharp e ns this bound on m ). Ravi A. Ra o has asked if the conv erse of the ab ov e is true ov er po lynomial rings R [ X ] with R loca l. Mor e precisely , if R is a lo cal ring with kR = R, do es K 0 ( R [ X ]) hav e non-trivial k- torsion? References [1] H. Bass; Al gebr aic K -the ory . W. A. Benjami n, Inc., New Y ork- Amsterdam (1968). [2] S. Blo ch; Al gebraic K-Theory and crystalli ne cohomology , Publ. Math. I.H.E.S. 47 (1977), 187–268. [3] S. Blo ch; Some formulas p ertaining to the K- theory of commutat ive group schemes, Journal of Algebr a 53 (1978), 304–326. [4] M. R. Gab el; Generic or thogonal stably free pro jectives, Journal of Algebr a 29 (1974), 477–488. [5] J. Gubeladze; Non tri viality of SK 1 ( R [ M ]), Journal of Pur e and Applie d Algebr a 104 (1995), 169–190. [6] J. Gubeladze; Classical algebraic K- theory of monoid algebras, K - the ory and homolo gic al algebr a (Tbilisi, 1987–88), 36–94, Lecture Notes in Math. 1437 (1990 ), Springer, Berlin. [7] S. Lang; Algebra, Third Edition. Pea rson Educ ati on Asia. [8] T.Y . Lam; Series summation of sta bly free mo dules, Q uart. J. Math. O xfor d Series (2) 27 (1976), no. 105, 37–46 . [9] P .M . Cohn; On the Structure of GL 2 of a Ring, Publ. Math. 30 (1966) , 5–54. [10] J. Stienstra; Op eration in the l inear K-theory of endomorphisms, Curr ent T r ends in Algebr aic T op olo g y , Conf. Pro c. Can. M ath. Soc. 2 (1982). [11] C. W eib el; May er-Vietoris Seque nce and mo dule str ucture on NK 0 , Springer L e ctur e Notes in Mathematics 854 (1981), 466–498. [12] C. W eib el; Module Structures on the K-theory of Graded Ri ngs, Journal of Algebr a 105 (1987), 465–483. Indian Instit ute of Scienc e Educ ation and R ese ar ch (IISER-K) , Mohanpur Campu s, P.O. BCKV Campus Main Offic e , Mohanpur, Nadia - 741252, West Bengal, India. F ax: 03473-2334-41 07 email : r ab eya.b asu@gmail.c om, rb asu@iiserkol.ac.in 6