aa r X i v : . [ m a t h . A C ] J u l ABSENCE OF TORSION IN ORBIT SPACE
SAMPAT SHARMA A BSTRACT . In this paper, we prove that if R is a local ring of dimension d, d ≥ and d ! ∈ R then the group Um d +1 ( R [ X ]) E d +1 ( R [ X ]) has no k -torsion, provided k ∈ GL ( R ) . We also prove that if R is a regular ring of dimension d, d ≥ and d ! ∈ R such that E d +1 ( R ) acts transitively on Um d +1 ( R ) then E d +1 ( R [ X ]) acts transitivelyon Um d +1 ( R [ X ]) .
1. I
NTRODUCTION
In [14], J. Stienstera, using the ideas of S. Bloch in [2] showed that
N K ( R ) is a W ( R ) -module where N K ( R ) = Ker ( K ( R [ X ]) −→ K ( R )); X = 0 and W ( R ) is the ring of big Witt vectors. Consequently,as noted by C. Weibel in [18], SK ( R [ X ]) has no k -torsion if k ∈ GL ( R ) and R is a commutative localring. Note that SK ( R [ X ]) coincides with N K ( R ) for a commutative local ring R . In [1], R. Basusimplified J. Stienstera’s approach of big Witt vector and proved that N K ( R ) has no k -torsion for anassociative ring R with kR = R. In ( [3, Theorem 2.2] ) , J. Fasel proved that for a non-singular affine algebra R of dimension d ≥ over a perfect field k of cohomological dimension atmost 1, the universal Mennicke symbol M S d +1 ( R ) isuniquely p divisible for p prime to the characteristic of k . Using results of J. Fasel ( [3, Theorem 2.1] ) , W.van der Kallen ( [10, Theorem 4.1] ) , Ravi Rao and Selby Jose in [8] concluded that for a non-singular affinealgebra R of dimension d ≥ over a perfect field k , char ( k ) = 2 , of cohomological dimension atmost 1,the orbit space Um d +1 ( R ) E d +1 ( R ) is uniquely p divisible for p prime to the characteristic of k .In ( [19, Corollary 7.4] ) , L.N Vaserstein proves that if R is a local ring of dimension , in which is invertible then Um ( R [ X ]) E ( R [ X ]) ≃ W E ( R [ X ]) . Using Karoubi’s linearization technique, in [13], Rao-Swanproved that W E ( R [ X ]) ֒ → SK ( R [ X ]) is an injective group homomorphism for a local ring R with R = R. Using Weibel’s result in [18],Ravi Rao and Richard Swan (for a proof see [8]), proved that Um ( R [ X ]) E ( R [ X ]) has no -torsion for a local ring of dimension with R = R. In this article, using the theory of Suslin matrices, we generalise the result of Ravi Rao and RichardSwan for any d -dimensional local ring, d ≥ . We prove that : Theorem 1.1.
Let R be a local ring of dimension d and let d ! ∈ R , then the group Um d +1 ( R [ X ]) E d +1 ( R [ X ]) has no k -torsion, provided k ∈ GL ( R ) . Using Ravi Rao’s result ( [12, Theorem 2.4] ) , we define a map φ : Um d +1 ( R [ X ]) E d +1 ( R [ X ]) −→ SK ( R [ X ]) .Weibel’s result that SK ( R [ X ]) has no k -torsion for a local ring R with k ∈ GL ( R ) , ensures that the map φ is well defined. Next we use a result of Anuradha Garge and Ravi Rao ( [4, Corollary 2.4] ) , to establishthat φ is an injective map. Using the fact that [ v ] [ S n ( v, w )] is a Mennicke symbol, one proves that φ isa group homomorphism.As an application of the Theorem 1.1, we answer a question raised by Ravi Rao and Selby Jose in [8].One can define an injective map from orbit space to the quotient group SO d +1) ( R [ X ]) /EO d +1) ( R [ X ]) . We prove that
Corollary 1.2.
Let R be a local ring of dimension d , d = 2 k, k ≥ , d ! ∈ R . Then the map ϕ : U m d +1 ( R [ X ]) E d +1 ( R [ X ]) −→ SO d +1) ( R [ X ]) EO d +1) ( R [ X ]) is injective. In [16], A. A. Suslin proved that for a noetherian ring of dimension d , E d +2 ( R [ X ]) acts transitively on U m d +2 ( R [ X ]) for d ≥ . As an application of Theorem 1.1, we improve this bound upto d + 1 in certainrings. We prove that : Corollary 1.3.
Let R be a regular ring of dimension d, d ≥ and d ! ∈ R such that E d +1 ( R ) actstransitively on U m d +1 ( R ) then E d +1 ( R [ X ]) acts transitively on U m d +1 ( R [ X ]) .
2. P
RELIMINARY R ESULTS
A row v = ( a , a , . . . , a r ) ∈ R r +1 is said to be unimodular if there is a w = ( b , b , . . . , b r ) ∈ R r +1 with h v, w i = Σ ri =0 a i b i = 1 and U m r +1 ( R ) will denote the set of unimodular rows (over R ) of length r + 1 .The group of elementary matrices is a subgroup of GL r +1 ( R ) , denoted by E r +1 ( R ) , and is generatedby the matrices of the form e ij ( λ ) = I r +1 + λE ij , where λ ∈ R, i = j, ≤ i, j ≤ r + 1 , E ij ∈ M r +1 ( R ) whose ij th entry is and all other entries are zero. The elementary linear group E r +1 ( R ) acts on the rowsof length r + 1 by right multiplication. Moreover, this action takes unimodular rows to unimodular rows : Um r +1 ( R ) E r +1 ( R ) will denote set of orbits of this action; and we shall denote by [ v ] the equivalence class of a row v under this equivalence relation. Definition 2.1. Essential dimension:
Let R be a ring whose maximal spectrum Max ( R ) is a finite unionof subsets V i , where each V i , when endowed with the (topology induced from the) Zariski topology is aspace of Krull dimension d. We shall say R is essentially of dimension d in such a case. Example 2.2.
Let R be a reduced local ring of dimension d ≥ , then R [ X ] is essentially of dimension d as for any non-zero-divisor π ∈ R, Spec ( R [ X ]) = Spec ( R ( π ) [ X ]) ∪ Spec ( R π [ X ]) is a finite union of noetherian spaces of dimension d. BSENCE OF TORSION IN ORBIT SPACE 3 In ( [9, Theorem 3.6] ) , W. van der Kallen derives an abelian group structure on Um d +1 ( R ) E d +1 ( R ) when R is ofessential dimension d, for all d ≥ . We will denote the group operation in this group by ∗ . Abelian group structure on Um n ( R ) E n ( R ) for sdim ( R ) ≤ n − .Definition 2.3. Stable range condition sr n ( R ) : We shall say stable range condition sr n ( R ) holds for R iffor any ( a , a , . . . , a n +1 ) ∈ U m n +1 ( R ) there exists c i ∈ R such that ( a + c a n +1 , a + c a n +1 , . . . , a n + c n a n +1 ) ∈ U m n ( R ) . Definition 2.4.
Stable range sr ( R ) , Stable dimension sdim ( R ) : We shall define the stable range of R denoted by sr ( R ) to be the least integer n such that sr n ( R ) holds. We shall define stable dimension of R by sdim ( R ) = sr ( R ) − . Definition 2.5.
Universal weak Mennicke symbol WMS n ( R ) , n ≥ We define the universal weakMennicke symbol on Um n ( R ) E n ( R ) by a set map wms : Um n ( R ) E n ( R ) −→ WMS n ( R ) , [ v ] wms ( v ) to a groupWMS n ( R ) . The group WMS n ( R ) is the free abelian group generated by wms ( v ) , v ∈ U m n ( R ) modulothe following relations • wms ( v ) = wms ( vε ) if ε ∈ E n ( R ) . • If ( q, v , . . . , v n ) , (1 + q, v , . . . , v n ) ∈ U m n ( R ) and r (1 + q ) ≡ q mod ( v , . . . , v n ) , then wms ( q, v , . . . , v n ) = wms ( r, v , . . . , v n ) wms (1 + q, v , . . . , v n ) . We recall ( [10, Theorem 4.1] ) . Theorem 2.6. (W. van der Kallen) Let R be a ring of stable dimension d, d ≤ n − and n ≥ . Then theuniversal weak Mennicke symbol wms : Um n ( R ) E n ( R ) −→ WMS n ( R ) is bijective. In view of above theorem Um n ( R ) E n ( R ) gets a group structure for d ≤ n − , n ≥ . We will denote groupoperation in Um n ( R ) E n ( R ) by ∗ . In ( [10, Lemma 3.5] ) , W. van der Kallen proves various revealing formulae inthis group. Let us recall some of these; • [ x , v , . . . , v n ] ∗ [ v , v , . . . , v n ] = [( v ( x + w ) − , ( x + w ) v , . . . , v n )] , where w is such that Σ ni =1 v i w i = 1 , for some w i ∈ R, ≤ i ≤ n. In particular [ v , v , . . . , v n ] − = [( − w , v , . . . , v n )] . • [ x , v , . . . , v n ] ∗ [ v , v , . . . , v n ] − = [(1 − w ( x − v ) , ( x − v ) v , . . . , v n )] . • [( x , v , . . . , v n )] ∗ [( v , v , . . . , v n )] = [( x v , v , . . . , v n )] . Mennicke–Newman Lemma.
We note Mennicke–Newman lemma proved by W. van der Kallen in ( [11, Lemma 3.2] ) : Lemma 2.7.
Let R be a commutative ring with sdim ( R ) ≤ n − . Let v, w ∈ U m n ( R ) . Then there exists ε, δ ∈ E n ( R ) and x, y, a i ∈ R such that vε = ( x, a , . . . , a n ) , wδ = ( y, a , . . . , a n ) , x + y = 1 . SAMPAT SHARMA
L.N Vaserstein’s power operation χ k on Um r ( R ) E r ( R ) , for k ∈ Z , r ≥ . In [20], L.N. Vaserstein hasshown that taking k-th power of a co-ordinate is a well defined operation χ k on Um r ( R ) E r ( R ) , r ≥ , for anycommutative ring R. Let E ∼ denote equivalence under the elementary group E r ( A ) . Thus, • ( v , . . . , v ki , . . . , v r ) E ∼ ( v , . . . , v kj , . . . , v r ) , for all ≤ i, j ≤ r, k ≥ . • ( v , . . . , v i − , w i , v i +1 , . . . , v r ) E ∼ ( v , . . . , v j − , w j , v j +1 , . . . , v r ) , for all ≤ i, j ≤ r, where Σ ri =1 v i w i = 1 , for some w i ∈ R. On the equality of χ k ([ v ]) with [ v ] k . W. van der Kallen in ( [10, Section 4] ) , gives an example toillustrate that, in general, ( v k , v , . . . , v n ) / ∈ [ v ] k , where v = ( v , . . . , v n ) . A cause for this anomaly waspointed out by Ravi A. Rao in [12] − Antipodal unimodular rows may not coincide up to an elementaryaction. Following ( [12, Lemma 1.3.1] ) , we prove the following : Lemma 2.1.
Let v = ( v , v , . . . , v n ) ∈ U m n ( R ) , where sdim ( R ) = d, d ≤ n − . Let us assume that v E ∼ ( − v , v , . . . , v n ) (equivalently, χ − ([ v ]) = [ v ] − ), then ( v k , v , . . . , v n ) ∈ [ v ] k for all k ∈ N . Proof : Let Σ ni =1 v i w i = 1 , for some w i ∈ R, ≤ i ≤ n. Note that we can then write v w +Σ ni =2 v i w ′ i = 1 for some w ′ i ∈ R, for ≤ i ≤ n. Now [( w , v , . . . , v n )] = χ − ([ v ]) = χ − ([ − v , . . . , v n ]) = [( − w , v , . . . , v n )] . Hence, [ v ] = [( − w , v , . . . , v n )] − = [( v w , v , . . . , v n )]= [( w , v , . . . , v n )] ∗ [( v , v , . . . , v n )]= [( − w , v , . . . , v n )] ∗ [( v , v , . . . , v n )]= [ v ] − ∗ [( v , v , . . . , v n )] . Therefore [ v ] = [( v , v , . . . , v n )] . It is now easy to check, via group identities in Um n ( R ) E n ( R ) , that [( v k , v , . . . , v n )] = [ v ] k , for all k > . (cid:3) The elementary symplectic Witt group W E ( R ) . If α ∈ M r ( R ) , β ∈ M s ( R ) are matrices then α ⊥ β denotes the matrix " α β ∈ M r + s ( R ) . ψ will denote " − ∈ E ( Z ) , and ψ r is inductivelydefined by ψ r = ψ r − ⊥ ψ ∈ E r ( Z ) , for r ≥ .A skew-symmetric matrix whose diagonal elements are zero is called an alternating matrix. If φ ∈ M r ( R ) is alternating then det ( φ ) = ( pf ( φ )) where pf is a polynomial (called the Pfaffian) in the matrixelements with coefficients ± . Note that we need to fix a sign in the choice of pf; so we insist pf ( ψ r ) = 1 for all r . For any α ∈ M r ( R ) and any alternating matrix φ ∈ M r ( R ) we have pf ( α t φα ) = pf ( φ ) det ( α ) .For alternating matrices φ, ψ it is easy to check that pf ( φ ⊥ ψ ) = ( pf ( φ ))( pf ( ψ )) .Two matrices α ∈ M r ( R ) , β ∈ M s ( R ) are said to be equivalent (w.r.t. E ( R ) ) if there exists a matrix ε ∈ SL r + s + l ) ( R ) T E ( R ) , such that α ⊥ ψ s + l = ε t ( β ⊥ ψ r + l ) ε, for some l . Denote this by α E ∼ β .Thus E ∼ is an equivalence relation; denote by [ α ] the orbit of α under this relation. BSENCE OF TORSION IN ORBIT SPACE 5
It is easy to see ( [19, p. 945] ) that ⊥ induces the structure of an abelian group on the set of all equivalenceclasses of alternating matrices with Pfaffian ; this group is called elementary symplectic Witt group and isdenoted by W E ( R ) .2.6. The Suslin matrices.
First recall the Suslin matrix S r ( v, w ) . These were defined by Suslin in ( [15,Section 5] ) . We recall his inductive process : Let v = ( a , v ) , w = ( b , w ) , where v , w ∈ M ,r ( R ) .Set S ( v, w ) = a and set S r ( v, w ) = " a I r − S r − ( v , w ) − S r − ( w , v ) T b I r − . The process is reversible and given a Suslin matrix S r ( v, w ) one can recover the associated rows v, w ,i.e. the pair ( v, w ) .In [17], Suslin proves that if h v, w i = v · w T = 1 , then by row and column operations one can reduce S r ( v, w ) to a matrix β r ( v, w ) ∈ SL r +1 ( R ) , whose first row is ( a , a , a · · · , a rr ) . This in particularproves that rows of such type can be completed to an invertible matrix of determinant one. We call any such β r ( v, w ) to be a compressed Suslin matrix.2.7. Relative elementary groups.Definition 2.2.
Let I be an ideal of a ring R. A unimodular row v ∈ U m n ( R ) which is congruent to e modulo I is called unimodular relative to ideal I. Set of unimodular rows relative to ideal I will be denotedby U m n ( R, I ) . Let I be an ideal of a ring R , we shall denote by GL n ( R, I ) the kernel of the canonical mapping GL n ( R ) −→ GL n (cid:0) RI (cid:1) . Let SL n ( R, I ) denotes the subgroup of GL n ( R, I ) consisting of elements ofdeterminant . Definition 2.3.
The relative groups E n ( I ) , E n ( R , I ) : Let I be an ideal of R . The elementary group E n ( I ) is the subgroup of E n ( R ) generated as a group by the elements e ij ( x ) , x ∈ I, ≤ i = j ≤ n. The relative elementary group E n ( R, I ) is the normal closure of E n ( I ) in E n ( R ) . Definition 2.4.
Excision ring : Let R be a ring and I be an ideal of R . The excision ring R ⊕ I , hascoordinate wise addition and multiplication is given as follows: ( r, i ) . ( s, j ) = ( rs, rj + si + ij ) , where r, s ∈ R and i, j ∈ I. The multiplicative identity of this group is (1 , and the additive identity is (0 , . Lemma 2.5. ( [5, Lemma 4.3] ) Let ( R, m ) be a local ring. Then the excision ring R ⊕ I with respect to aproper ideal I ( R is also a local ring with maximal ideal m ⊕ I . Definition 2.6.
We shall say a ring homomorphism φ : B ։ D has a section if there exists a ringhomomorphism γ : D ֒ → B so that φ ◦ γ is the identity on D. We shall also say D is a retract of B. The following lemma is an easy consequence of ( [6, Lemma 4.3, Chapter 3] ) : SAMPAT SHARMA
Lemma 2.7. ( A. Suslin ) Let
B, D be rings and π : B ։ D have a section. If J = ker ( π ) , then E n ( B, J ) = E n ( B ) ∩ SL n ( B, J ) , n ≥ . Remark 2.8.
Let R be a ring and I be an ideal of R . There is a natural homomorphism ω : R ⊕ I → R given by ( x, i ) x + i ∈ R. Clearly ω has a section. Let v = (1 + i , i , . . . , i n ) ∈ U m n ( R, I ) where i j ’sare in I. Then we shall call ˜ v = ((1 , i ) , (0 , i ) , . . . , (0 , i n )) ∈ U m n ( R ⊕ I, ⊕ I ) to be a lift of v. Notethat ω sends ˜ v to v.
3. A
BSENCE OF TORSION IN Um d +1 ( R [ X ]) E d +1 ( R [ X ]) In ( [19, Corollary 7.4] ) , L.N Vaserstein proves that if R is a local ring of dimension , in which isinvertible then Um ( R [ X ]) E ( R [ X ]) ≃ W E ( R [ X ]) . Since, W E ( R [ X ]) ֒ → SK ( R [ X ]) (See Corollary 3.3) for alocal ring R with R = R, one gets Um ( R [ X ]) E ( R [ X ]) has no -torsion for a local ring of dimension . In thissection we generalise this result for any d -dimension local ring, d ≥ , via a different approach.We first collect some known results which will be used in this section. Proposition 3.1. ( [18, Section 3] ) Let R be a local ring. Assume that mR = R . Then SK ( R [ X ]) has no m -torsion. Next we recall an observation of Rao-Swan in [13], about a result of Karoubi, and its consequence,whose proofs can be found in ( [4, Lemma 4.2, Corollary 4.3] ) . Lemma 3.2. (Karoubi)
Let R be a commutative ring with . Let φ + nX , with φ ∈ GL k ( R ) , n ∈ M k ( R ) , be a linear invertible polynomial alternating matrix over R . If R = R , then φ + nX = W ( X ) t ( φ − ) t W ( X ) , where W ( X ) = φ (1 + φ − nX ) . Corollary 3.3. (Rao-Swan)
Let R be a local ring with , in which R = R . Then the natural map W E ( R [ X ]) −→ SK ( R [ X ]) is an injective group homomorphism. Proof : For a proof see ( [4, Corollary 4.3] ) . (cid:3) Corollary 3.4.
Let R be a local ring and mR = R . Then W E ( R [ X ]) does not have any m -torsion. Proof : It follows from Proposition 3.1 and Corollary 3.3. (cid:3)
Theorem 3.5. ( [12, Theorem 2.4] ) Let R be a local ring of dimension d ≥ . Let v ∈ U m d +1 ( R [ X ]) . If d ! ∈ R then v E ∼ ( w d !0 , w , . . . , w d ) for some ( w , . . . , w d ) ∈ U m d +1 ( R [ X ]) . Remark 3.6.
In general, if R is a local ring of dimension d, d ≥ , k ∈ R and v ∈ U m d +1 ( R [ X ]) . Thenone can similarly prove that v E d +1 ( R [ X ]) ∼ ( u k , u , . . . , u d +1 ) for some ( u , . . . , u d +1 ) ∈ U m d +1 ( R [ X ]) . Theorem 3.7.
Let R be a commutative ring with , sdim ( R ) ≤ n − , n ! ∈ R and n ≥ . Assume that (1)
For all v ∈ U m n +1 ( R ) , v E n +1 ( R ) ∼ ( w n !0 , w , . . . , w n ) . BSENCE OF TORSION IN ORBIT SPACE 7 (2) If β n ( v, w ) ∈ SL n +1 ( R ) ∩ E ( R ) , then [ e β n ( v, w )] = [ e ] . (3) SK ( R ) has no k -torsion, if k ∈ R. Then, for some w with h e σ, w i = 1 , the map ϕ : U m n +1 ( R ) E n +1 ( R ) −→ SK ( R )[ v ] = [ e σ ] n ! [ S n ( e σ, w )] is a well-defined injective group homomorphism. Proof : Well-defined:
In view of the hypothesis (1) , for any v ∈ U m n +1 ( R ) , there exists v ′ ∈ U m n +1 ( R ) , such that [ v ] = χ n ! ([ v ′ ]) . Since every v ′ ∈ U m n +1 ( R ) is completable, so by Lemma 2.1, [ v ] = [ e σ ] n ! , for some σ ∈ SL n +1 ( R ) . If v ′ , v ′′ ∈ U m n +1 ( R ) be such that [ v ] = [ v ′ ] n ! = [ v ′′ ] n ! . By hypothesis (1) , there exists τ, τ ′ ∈ SL n +1 ( R ) such that [ v ] = [ e τ ] n ! = [ e τ ′ ] n ! . By ( [7, Lemma 3.2] ) , [ S n ( e τ n ! , w ′ )] = [ S n ( e τ ′ n ! , w ′′ )] . Since [ v ] [ S n ( v, w )] is a Mennicke symbol and using Lemma 2.1, we have [ S n ( e τ, w ′ )] n ! = [ S n ( e τ ′ , w ′′ )] n ! . Since SK ( R ) does not have any ( n !) -torsion, we have [ S n ( e τ, w ′ )] = [ S n ( e τ ′ , w ′′ )] . Therefore the map ϕ : U m n +1 ( R ) E n +1 ( R ) −→ SK ( R )[ v ] = [ e σ ] n ! [ S n ( e σ, w )] for some w ∈ U m n +1 ( R ) with ( e σ ) · w t = 1 , is well-defined. Injectivity:
Let v = [ e σ ] n ! be such that [ S n ( e σ, w )] = 1 in SK ( R ) . By ( [21, Theorem 3.2] ) , β n ( e σ, w ) ∈ SL n +1 ( R ) ∩ E ( R ) . By hypothesis (2) , [ v ] = [ e σ ] n ! = [ e ] . Thus φ is injective. Homomorphism:
Let v, w ∈ U m n +1 ( R ) be such that [ v ] = [ e σ ] n ! and [ w ] = [ e τ ] n ! for somematrices σ, τ ∈ SL n +1 ( R ) . In view of Mennicke-Newman lemma (Lemma 2.7), we may assume that [ e σ ] = [ a, x , x , · · · , x n ] and [ e τ ] = [ b, x , x , · · · , x n ] . Now, in view of Lemma 2.1, we have [ e σ ] n ! =[ a n ! , x , x , · · · , x n ] and [ e τ ] n ! = [ b n ! , x , x , · · · , x n ] . Thus by ( [9, Theorem 3.16(iii)] ) , we have [ v ] ∗ [ w ] = [( ab ) n ! , x , x , · · · , x n ] . Now, by hypothesis (1) , there exists τ ′ ∈ SL n +1 ( R ) be such that [ e τ ′ ] = [ ab, x , x , · · · , x n ] . Thus, ϕ ([ v ] ∗ [ w ]) = [ S n ( e τ ′ , w ′ )]= [ S n (( ab, x , . . . , x n ) , w ′ )] . Since v S n ( v, w ) is a Mennicke symbol, we have ϕ ([ v ] ∗ [ w ]) = [ S n (( ab, x , . . . , x n ) , w ′ )]= [ S n (( a, x , . . . , x n ) , w )][ S n (( b, x , . . . , x n ) , w )]= [ S n ( e σ, w )][ S n ( e τ, w )]= ϕ ([ v ]) ϕ ([ w ]) . (cid:3) Corollary 3.8.
Let R be a commutative ring with , sdim ( R ) ≤ n − and n ≥ . If SAMPAT SHARMA (1)
For all v ∈ U m n +1 ( R ) , v E n +1 ( R ) ∼ ( w n !0 , w , . . . , w n ) . (2) If β n ( v, w ) ∈ SL n +1 ( R ) ∩ E ( R ) , then [ e β n ( v, w )] = [ e ] . (3) SK ( R ) has no k -torsion if k ∈ R. Then the group Um n +1 ( R ) E n +1 ( R ) has no k -torsion. Proof : In view of Theorem 3.7,
U m n +1 ( R ) E n +1 ( R ) ֒ → SK ( R ) is an injection. Since SK ( R ) does not have any k -torsion, Um n +1 ( R ) E n +1 ( R ) has no k -torsion. (cid:3) Corollary 3.9.
Let R be a local ring of dimension d, d ≥ and let d ! ∈ R , then Um d +1 ( R [ X ]) E d +1 ( R [ X ]) has no k -torsion, provided k ∈ GL ( R ) . Proof : It follows from Corollary 3.8 as all the hypothesis are satisfied in view of Theorem 3.5, ( [4,Corollary 2.2] ) , and Proposition 3.1 respectively. (cid:3) Lemma 3.10.
Let R be a local ring of dimension d ≥ , d ! ∈ R and I be a proper ideal in R. Let v ∈ U m d +1 ( R [ X ] , I [ X ]) . Then Um d +1 ( R [ X ] ,I [ X ]) E d +1 ( R [ X ] ,I [ X ]) has no k -torsion, provided k ∈ GL ( R ) . Proof : Let us assume that [ v ] k E d +1 ( R [ X ] ,I [ X ]) ∼ e . Let ˜ v ∈ U m d +1 (( R ⊕ I )[ X ] , (0 ⊕ I )[ X ]) . In viewof Lemma 2.5 and ( [9, Lemma 3.19] ) , R ⊕ I is a local ring of dimension d. By Corollary 3.9, the group Um d +1 (( R ⊕ I )[ X ]) E d +1 (( R ⊕ I )[ X ]) has no k -torsion. Thus there exists ε ∈ E d +1 (( R ⊕ I )[ X ]) such that ˜ vε = e . Goingmodulo ⊕ I, we have e − ε = e , − ε ∈ SL d +1 ( R [ X ]) . Now replacing ε by ε ( − ε ) − and using Lemma 2.7,we may assume that ε ∈ E d +1 (( R ⊕ I )[ X ] , (0 ⊕ I )[ X ]) satisfying ˜ vε = e . Now applying ω to last equationwe get vε = e for some ε ∈ E d +1 ( R [ X ] , I [ X ]) which proves that Um d +1 ( R [ X ] ,I [ X ]) E d +1 ( R [ X ] ,I [ X ]) has no k -torsion. (cid:3) Corollary 3.11.
Let R be a local ring of dimension d , d = 2 k, k ≥ , d ! ∈ R . Then the map ϕ : U m d +1 ( R [ X ]) E d +1 ( R [ X ]) −→ SO d +1) ( R [ X ]) EO d +1) ( R [ X ]) is injective. Proof : Let [ v ] ∈ Um d +1 ( R [ X ]) E d +1 ( R [ X ]) such that ϕ ([ v ]) ∈ EO d +1) ( R [ X ]) . In view of ( [8, Theorem 3.8] ) ,χ [ v ] = e . Now, we use Corollary 3.9 to conculde that [ v ] = e , i.e. ϕ is injective. (cid:3)
4. A
N APPLICATION TO COMPLETION OF UNIMODULAR ROWS In ( [16, Theorem 2.6] ) , A. A. Suslin proved the following result:
Theorem 4.1. ( A.A. Suslin ) Let R be a noetherian ring of dimension d , then E r ( R [ X ]) acts transitively on U m r ( R [ X ]) for r ≥ Max (3 , d + 2) . As an application of Theorem 3.7, we improve the bound to d + 1 over regular local rings. We prove : BSENCE OF TORSION IN ORBIT SPACE 9
Corollary 4.2.
Let R be a regular local ring of dimension d, d ≥ and let d ! ∈ R , then E r ( R [ X ]) actstransitively on U m r ( R [ X ]) for r ≥ Max (3 , d + 1) . Proof : If r ≥ d + 2 then it follows from Suslin’s theorem. Let r = d + 1 . In view of theorem 3.7, thereis an injective group homomorphism ϕ : U m d +1 ( R [ X ]) E d +1 ( R [ X ]) −→ SK ( R [ X ]) . Since R is a regular local ring SK ( R [ X ]) = 0 . Thus E d +1 ( R [ X ]) acts transitively on U m d +1 ( R [ X ]) . (cid:3) Corollary 4.3.
Let R be a regular ring of dimension d, d ≥ , d ! ∈ R such that E d +1 ( R ) acts transitivelyon U m d +1 ( R ) , then E r ( R [ X ]) acts transitively on U m r ( R [ X ]) for r ≥ Max (3 , d + 1) . Proof : If r ≥ d + 2 then it follows from Suslin’s theorem. Let r = d + 1 . For any maximal ideal m of R , in view of corollary 4.2, E d +1 ( R m [ X ]) acts transitively on U m d +1 ( R m [ X ]) . By Sulsin’s local-globalprinciple, we gets v ( X ) ∼ E d +1 ( R [ X ]) v (0) for every v ( X ) ∈ U m d +1 ( R [ X ]) . Now result follows from ourassumption. (cid:3)
Corollary 4.4.
Let R be a regular finitely generated ring over Z of dimension d, d ≥ , d ! ∈ R. Then E d +1 ( R [ X ]) acts transitively on U m d +1 ( R [ X ]) . Proof : The result follows in view of ( [19, Theorem 18.2] ) and corollary 4.3. (cid:3) Corollary 4.5.
Let R be a regular finitely generated algebra over a field K of dimension d, d ≥ , d ! ∈ R. Then E d +1 ( R [ X ]) acts transitively on U m d +1 ( R [ X ]) . Proof : The result follows in view of ( [19, Theorem 20.5] ) and corollary 4.3. (cid:3) In ( [16, Corollary 2.7] ) , A.A. Suslin proved the following result:
Corollary 4.6. ( A.A. Suslin ) Let R be a noetherian ring of dimension d , then the canonical mapping ϕ : GL r ( R [ X ]) −→ K ( R [ X ]) is an epimorphism for r ≥ d + 1 . As an application of corollary 4.3, we prove :
Corollary 4.7. ( A.A. Suslin ) Let R be a regular ring of dimension d, d ≥ , d ! ∈ R such that E d +1 ( R ) acts transitively on U m d +1 ( R ) ,, then the canonical mapping ϕ : GL r ( R [ X ]) −→ K ( R [ X ]) is an epimorphism for r ≥ d + 1 . Proof : The result is obvious in view of corollary 4.3. (cid:3) R EFERENCES[1] R. Basu;
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AMPAT S HARMA , S
CHOOL OF M ATHEMATICS , T
ATA I NSTITUTE OF F UNDAMENTAL R ESEARCH ,1, D R . H OMI B HABHA R OAD , M
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