aa r X i v : . [ m a t h . A C ] J a n COMPLETION OF SKEW COMPLETABLE UNIMODULAR ROWS
SAMPAT SHARMA A BSTRACT . In this paper, we prove that if R is a local ring of dimension d ≥ , d odd and d − ∈ R then any skew completable unimodular row v ∈ Um d ( R [ X ]) is completable. It is also proved that skew completable unimodularrows of size d ≥ over a regular local ring of dimension d are first row of a 2-stably elementary matrix. Throughout this article we will assume R to be a commutative noetherian ringwith = 0 .
1. I
NTRODUCTION
In 1955, J.P. Serre asked whether there were non-free projective modules overa polynomial extension k [ X , . . . , X n ] , over a field k . D. Quillen ([6]) and A.A.Suslin ([9]) settled this problem independently in early 1976; and is now known asthe Quillen–Suslin theorem. Since every finitely generated projective module over k [ X , . . . , X n ] is stably free, to determine whether projective modules are free, itis enough to determine that unimodular rows over k [ X , . . . , X n ] are completable.Therefore, problem of completion of unimodular rows is a central problem inclassical K -Theory.In [11], R.G. Swan and J. Towber showed that if ( a , b, c ) ∈ U m ( R ) then it canbe completed to an invertible matrix over R. This result of Swan and Towber wasgeneralised by Suslin in [10] who showed that if ( a r !0 , a , . . . , a r ) ∈ U m r +1 ( R ) then it can be completed to an invertible matrix. In [7], Ravi Rao studied theproblem of completion of unimodular rows over R [ X ] , where R is a local ring.Ravi Rao showed that if R is a local ring of dimension d ≥ , d ! ∈ R , then anyunimodular row over R [ X ] of length d + 1 can be mapped to a factorial row byelementary transformations. In [8], Ravi Rao proved that if R is a local ring ofdimension with R = R , then unimodular rows of length are completable. In[2], Ravi Rao generalised his result with Anuradha Garge and proved that if R is a local ring of dimension with R = R then any unimodular row of length canbe mapped to a factorial row via a two stably elementary matrix.In this article, we generalise the result of Garge–Rao for skew completableunimodular rows. We prove : Theorem 1.1.
Let R be a local ring of Krull dimension d ≥ with d odd and d − ∈ R. Let v = ( v , v , . . . , v d − ) ∈ U m d ( R [ X ]) be skew-completableunimodular row over R [ X ] . Then there exists ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) and an invertible alternating matrix W ∈ SL d +1 ( R [ X ]) such that vρ = e K ( W ) . In the last section, we study the completion of skew completable unimodularrows over regular local rings. Since SK ( R [ X ]) is trivial for a regular local ring R , we get the following result : Theorem 1.2.
Let R be a regular local ring of Krull dimension d ≥ with d oddand d − ∈ R. Let v = ( v , v , . . . , v d − ) ∈ U m d ( R [ X ]) be skew-completableunimodular row over R [ X ] . Then there exists ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) suchthat v = e ρ.
2. P
RELIMINARY R EMARKS
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 denotethe 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 generated by 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 andall other entries are zero. The elementary linear group E r +1 ( R ) acts on the rows oflength r + 1 by right multiplication. Moreover, this action takes unimodular rowsto unimodular rows : Um r +1 ( R ) E r +1 ( R ) will denote set of orbits of this action; and we shalldenote by [ v ] the equivalence class of a row v under this equivalence relation.2.1. 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 inductively defined by ψ r = ψ r − ⊥ ψ ∈ E r ( Z ) ,for r ≥ .A skew-symmetric matrix whose diagonal elements are zero is called an alter-nating matrix. If φ ∈ M r ( R ) is alternating then det ( φ ) = ( pf ( φ )) where pfis a polynomial (called the Pfaffian) in the matrix elements with coefficients ± . OMPLETION OF SKEW COMPLETABLE UNIMODULAR ROWS 3
Note that we need to fix a sign in the choice of pf; so we insist pf ( ψ r ) = 1 forall r . For any α ∈ M r ( R ) and any alternating matrix φ ∈ M r ( R ) we havepf ( α t φα ) = pf ( φ ) det ( α ) . For alternating matrices φ, ψ it is easy to check thatpf ( 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.It is easy to see ( [12, p. 945] ) that ⊥ induces the structure of an abelian group onthe set of all equivalence classes of alternating matrices with pfaffian ; this groupis called elementary symplectic Witt group and is denoted by W E ( R ) .2.2. W. Van der Kallen’s group structure on
U m d +1 ( R ) /E d +1 ( R ) .Definition 2.1. Essential dimension: Let R be a ring whose maximal spectrumMax ( R ) is a finite union of subsets V i , where each V i , when endowed with the(topology induced from the) Zariski topology is a space of Krull dimension d. Weshall say R is essentially of dimension d in such a case.For instance, a ring of Krull dimension d is obviously essentially of dimension ≤ d ; a local ring of dimension d is essentially of dimension whereas a polynomialextension R [ X ] of a local ring R of dimension d ≥ has dimension d + 1 but isessentially of dimension d as Max ( R [ X ]) = Max ( R/ ( a )[ X ]) ∪ Max ( R a [ X ]) forany non-zero divisor a ∈ R. In ( [4, Theorem 3.6] ) , W. van der Kallen derives an abelian group structure on Um d +1 ( R ) E d +1 ( R ) when R is essentially of dimension d, for all d ≥ . Let ∗ denote thegroup multiplication henceforth. He also proved in ( [4, Theorem 3.16(iv)] ) , thatthe first row map is a group homomorphism SL d +1 ( R ) −→ U m d +1 ( R ) E d +1 ( R ) when R is essentially of dimension d, for all d ≥ . Lemma 2.2.
Let R be essentially of dimension d ≥ , and let C d +1 ( R ) denote theset of all completable ( d + 1) -rows in U m d +1 ( R ) . Then, • The map σ −→ [ e σ ] , where e = (1 , , . . . , ∈ U m d +1 ( R ) , is a grouphomomorphism SL d +1 ( R ) −→ Um d +1 ( R ) E d +1 ( R ) . • C d +1 ( R ) E d +1 ( R ) is a subgroup of Um d +1 ( R ) E d +1 ( R ) . Proof : First follows from ( [4, Theorem 3.16(iv)] ) . Since v ∈ C d +1 ( R ) canbe completed to a matrix of determinant one, C d +1 ( R ) E d +1 ( R ) is the image of SL d +1 ( R ) under the above mentioned homomorphism; whence is a subgroup of Um d +1 ( R ) E d +1 ( R ) . SAMPAT SHARMA
Proposition 2.3.
Let R be a local ring of dimension d , d ≥ and d − ∈ R. Let v = ( v , . . . , v d ) ∈ U m d +1 ( R [ X ]) . Then v is completable if and only if v ( d − =( v ( d − , v , . . . , v d ) is completable. Proof : In view of ( [7, Remark 1.4.3] ) , we may assume that R is a reducedring. By ( [7, Lemma 1.3.1, Example 1.5.3] ) , [ v ( d − ] = [ v ] ∗ [ v ] ∗ · · · ∗ [ v ] , ( d − timesin Um d +1 ( R [ X ]) E d +1 ( R [ X ]) . By Lemma 2.2, v is completable implies v ( d − is also com-pletable.Conversely, let v ( d − be completable. By ( [7, Proposition 1.4.4] ) ,v E ∼ ( w , w , . . . , w d − , c ) with c ∈ R a non-zero-divisor. Since dim ( R/ ( c )) = d − and d − ∈ R, by ( [7,Corollary 2.3] ) , ( − w , − w , . . . , − w d − ) ∈ e SL d ( R/ ( c )[ X ]) . By ( [3, Proposition 1.2, Chapter 5] ) , ( w , w , . . . , w d − , c d ) is completable.Thus, • ( v , v , . . . , v d − , v dd ) E ∼ ( w , w , . . . , w d − , c d ) by ( [14, Theorem 1] ) , • [ v ] n = [( v , v , . . . , v d − , v nd )] for all n by ( [7, Lemma 1.3.1] ) . Thus [ v ] d = [( w , w , . . . , w d − , c d )] ∈ C d +1 ( R [ X ]) E d +1 ( R [ X ]) and by hypothesis [ v ] d − =[ v ( d − ] ∈ C d +1 ( R [ X ]) E d +1 ( R [ X ]) . Therefore by Lemma 2.2, v is completable. (cid:3)
3. K
RUSEMEYER ’ S COMPLETION OF THE SQUARE OF A SKEW COMPLETABLEROW
Definition 3.1.
A row v ∈ U m r − ( R ) is said to be skew completable if there isan invertible alternating matrix V ∈ GL r ( R ) with e V = (0 , v ) . First we note an example of skew completable unimodular row which is notcompletable.
Example 3.2 (Kaplansky) . Let A = R [ x ,x ,x ]( x + x + x − and v = ( x , x , x ) ∈ U m ( A ) . In view of ( [12, Section 5] ) , every unimodular row of length 3 is skew completable.Thus v = ( x , x , x ) is skew completable. Next we will show that v is notcompletable.Suppose to the contrary that v = e σ for some σ ∈ SL ( A ) . Let σ = ( σ ij ) . Wecan think σ ij ’s as a function on S . Let us define tangent vector field φ : S −→ R w (( σ − ) t ( w ) , σ − ) t ( w ) , σ − ) t ( w )) . OMPLETION OF SKEW COMPLETABLE UNIMODULAR ROWS 5 As σ ij ’s are polynomials, φ is a differential function. Since ( σ − ) t ∈ SL ( A ) , φ is a nonvanishing continuous tangent vector field on S which is a contradiction toHairy ball theorem. Thus v is not completable. Theorem 3.3. ( M. Krusemeyer ) ( [5, Theorem 2.1] ) Let R be a commutative ringand v = ( v , . . . , v n ) be skew completable. Let V be a skew completion of v , then ( v , v , . . . , v n ) is completable. Notation 3.4.
In the above theorem we will denote K ( V ) ∈ SL n ( R ) to be a com-pletion of ( v , v , . . . , v n ) for a skew completable unimodular row v = ( v , . . . , v n ) and its skew completion V. Remark 3.5.
M. Krusemeyer’s proof in ( [5, Theorem 2.1] ) , shows that V ∈ (1 ⊥ K ( V )) E n +1 ( R ) . Lemma 3.6.
Let R be a commutative ring and v = ( v , . . . , v n ) ∈ U m n ( R ) beskew completable to V. Then [ e K ( V )] = [ e K ( V ) t ] . Proof : By Remark 3.5, V ∈ (1 ⊥ K ( V )) E n +1 ( R ) . Since − I k ∈ E k ( R ) , we have V ∈ V t E n +1 ( R ) . Therefore (1 ⊥ K ( V )) t ∈ (1 ⊥ K ( V )) E n +1 ( R ) . Since stably K ( V ) and K ( V ) t are in same elementary class, therefore in view of ( [13, Lemma 10] ) , we have [ e K ( V )] = [ e K ( V ) t ] . (cid:3) Lemma 3.7.
Let R be a local ring with / ∈ R and let V be an invertiblealternating matrix of Pfaffian . Let e V = (0 , v , . . . , v r − ) . Then [ V n ] =[ W ] , with e W = (0 , v n , . . . , v r − ) . Proof : We will prove it by induction on n. For n = 1 , by ( [2, Corollary 4.3] ) ,W E ( R [ X ]) ֒ → SK ( R [ X ]) is injective, we have V ⊥ V SK ≡ V SK ≡ V t ψ r V SK ≡ (1 ⊥ K ( V ) t ) ψ r (1 ⊥ K ( V )) . Therefore [ V ] = [ U ] with e U = (0 , v , . . . , v r − ) . Now assume that result istrue for all k ≤ n − and Let [ W ] = [ V n − ] with e W = (0 , v n − , . . . , v r − ) . Since by lemma ( [2, Corollary 4.3] ) , W E ( R [ X ]) ֒ → SK ( R [ X ]) is injective, wehave W ⊥ W SK ≡ W SK ≡ W t ψ r W SK ≡ (1 ⊥ K ( W ) t ) ψ r (1 ⊥ K ( W )) . Therefore [ V n ] = [ W ] with e W = (0 , v n , . . . , v r − ) . (cid:3)
4. C
OMPLETION OF SKEW - COMPLETABLE UNIMODULAR ROWS OF LENGTH d In this section, we prove that if R is a local ring of dimension d ≥ , d oddand d − ∈ R then any skew completable unimodular row v ∈ U m d ( R [ X ]) iscompletable. SAMPAT SHARMA
Proposition 4.1.
Let R be a local ring of dimension d ≥ with d odd and d − ∈ R. Let V ∈ SL d +1 ( R [ X ]) be an alternating matrix with Pfaffian . Then [ V ] =[(1 ⊥ K ( W ) t ) ψ d +12 (1 ⊥ K ( W ))] for some [ W ] ∈ W E ( R [ X ]) . Consequently,there is a 1-stably elementary matrix γ ∈ SL d +1 ( R [ X ]) such that V = γ t (1 ⊥ K ( W ) t ) ψ d +12 (1 ⊥ K ( W )) γ. Proof : By ( [8, Proposition 2.4.1] ) , [ V ] = [ W ] for some W ∈ W E ( R [ X ]) . By ( [9, Theorem 2.6] ) , U m r ( R [ X ]) = e E r ( R [ X ]) for r ≥ d + 2 , so on applying ( [12, Lemma 5.3 and Lemma 5.5] ) , a few times, if necessary, we can find analternating matrix W ∈ SL d +1 ( R [ X ]) such that [ W ] = [ W ] . Therefore [ V ] =[ W ] . Now, we have W ⊥ W SK ≡ W SK ≡ W t ψ d +12 W SK ≡ (1 ⊥ K ( W ) t ) ψ d +12 (1 ⊥ K ( W )) . Since in view of ( [2, Corollary 4.3] ) , W E ( R [ X ]) ֒ → SK ( R [ X ]) is injective.Thus [ W ] = [(1 ⊥ K ( W ) t ) ψ d +12 (1 ⊥ K ( W ))] . Therefore, [ V ] = [(1 ⊥ K ( W ) t ) ψ d +12 (1 ⊥ K ( W ))] . The last statement follows by applying ( [12, Lemma5.5 and Lemma 5.6] ) . (cid:3) Theorem 4.2.
Let R be a local ring of Krull dimension d ≥ with d odd and d − ∈ R. Let v = ( v , v , . . . , v d − ) ∈ U m d ( R [ X ]) be skew-completableunimodular row over R [ X ] . Then there exists ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) and an invertible alternating matrix W ∈ SL d +1 ( R [ X ]) such that vρ = e K ( W ) . Proof : Let V ∈ SL d +1 ( R [ X ]) be an invertible alternating matrix of Pfaffian which is a skew completion of v = ( v , v , . . . , v d − ) . By Proposition 4.1, thereexists an alternating matrix W ∈ SL d +1 ( R [ X ]) of Pfaffian such that [ V ] = [(1 ⊥ K ( W )) t ψ d +12 (1 ⊥ K ( W ))] . Therefore there exists γ ∈ SL d +1 ( R [ X ]) ∩ E d +2 ( R [ X ]) such that γ t V γ = (1 ⊥ K ( W )) t ψ d +12 (1 ⊥ K ( W )) . In view of ( [2, Corollary 5.17] ) , e γ can be completed to an elementary matrix.Thus there exists ε ∈ E d +1 ( R [ X ]) , and ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) such that ε t (1 ⊥ ρ ) t V (1 ⊥ ρ ) ε = (1 ⊥ K ( W )) t ψ d +12 (1 ⊥ K ( W )) . By ( [1, Corollary 4.5] ) , there exists ε ∈ E d ( R [ X ]) such that (1 ⊥ ε ) t (1 ⊥ ρ ) t V (1 ⊥ ρ )(1 ⊥ ε ) = (1 ⊥ K ( W )) t ψ d +12 (1 ⊥ K ( W )) . Now we set ρ = ρ ε . Thus vρ = e K ( W ) . Hence v is completable . (cid:3) OMPLETION OF SKEW COMPLETABLE UNIMODULAR ROWS 7
5. C
OMPLETION OF UNIMODULAR VECTORS ( [8, Theorem 3.1] ) , Ravi A. Rao proved that for a local ring of dimension , every v ∈ U m ( R [ X ]) is completable. We get stronger results than Theorem 4.2,when we work with a local ring R of dimension . We reprove Anuradha Gargeand Ravi Rao’s result in ( [2, Corollary 5.18] ) . Proposition 5.1.
Let R be a local ring of dimension with k ∈ R and let V ∈ SL ( R [ X ]) be an alternating matrix of Pfaffian . Then [ V ] = [ V ∗ ] in W E ( R [ X ]) with e V ∗ = (0 , a k , b, c ) , and V ∗ ∈ SL ( R [ X ]) . Consequently, there is a stablyelementary γ ∈ SL ( R [ X ]) such that V = γ t V ∗ γ. Proof : By ( [8, Proposition 2.4.1] ) , [ V ] = [ W ] k for some W ∈ W E ( R [ X ]) . By ( [9, Theorem 2.6] ) , U m r ( R [ X ]) = e E r ( R [ X ]) for r ≥ , so on applying ( [12, Lemma 5.3 and Lemma 5.5] ) , a few times, if necessary, we can find analternating matrix V ∗ ∈ SL ( R [ X ]) such that [ W ] = [ V ∗ ] . Therefore [ V ] =[ V ∗ ] k . Let [ V ∗ ] k = [ V ∗ ] , thus [ V ] = [ V ∗ ] . By ( [2, Lemma 4.8] ) , e V ∗ =(0 , a k , b, c ) . The last statement follows by applying ( [12, Lemma 5.3 and Lemma5.5] ) . (cid:3) Theorem 5.2.
Let R be a local ring of Krull dimension with k ∈ R. Let v =( v , v , v ) ∈ U m ( R [ X ]) . Then there exists ρ ∈ SL ( R [ X ]) ∩ E ( R [ X ]) suchthat vρ = ( a k , b, c ) for some ( a, b, c ) ∈ U m ( R [ X ]) . Proof : Choose w = ( w , w , w ) such that Σ i =0 v i w i = 1 , and consider thealternating matrix V with Pfaffian given by V = v v v − v w − w − v − w w − v w − w ∈ SL ( R [ X ]) . By Proposition 5.1, there exists an alternating matrix V ∗ ∈ SL ( R [ X ]) , with e V ∗ = (0 , a k , b, c ) , of Pfaffian such that [ V ] = [ V ∗ ] . Therefore there exists γ ∈ SL ( R [ X ]) ∩ E ( R [ X ]) such that γ t V γ = V ∗ . In view of ( [2, Corollary 5.17] ) , e γ can be completed to an elementary matrix.Thus there exists ε ∈ E ( R [ X ]) , and ρ ∈ SL ( R [ X ]) ∩ E ( R [ X ]) such that ε t (1 ⊥ ρ ) t V (1 ⊥ ρ ) ε = V ∗ . SAMPAT SHARMA By ( [1, Corollary 4.5] ) , there exists ε ∈ E ( R [ X ]) such that (1 ⊥ ε ) t (1 ⊥ ρ ) t V (1 ⊥ ρ )(1 ⊥ ε ) = V ∗ . Now we set ρ = ρ ε . Thus vρ = ( a k , b, c ) for some ( a, b, c ) ∈ U m ( R [ X ]) . (cid:3)
6. C
OMPLETION OVER REGULAR LOCAL RINGS
In this section, we prove that skew completable unimodular rows of size d ≥ over a regular local ring of dimension d are first row of a 2- stably elementarymatrix.We note a result of Rao and Garge in ( [2, Corollary 4.3] ) . Lemma 6.1.
Let R be a local ring with R = R. Then the natural map W E ( R [ X ]) ֒ → SK ( R [ X ]) is injective. Corollary 6.2.
Let R be a regular local ring with R = R. Then the Witt group W E ( R [ X ]) is trivial. Proof : Since R is a regular local ring, SK ( R [ X ]) = 0 . Thus the resultfollows in view of Lemma 6.1. (cid:3)
Lemma 6.3.
Let R be a regular local ring of Krull dimension d ≥ with d oddand d − ∈ R. Let v = ( v , v , . . . , v d − ) ∈ U m d ( R [ X ]) be skew-completableunimodular row over R [ X ] . Then there exists ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) suchthat v = e ρ. Proof : Let V be a skew completion of v. In view of Corollary 6.2, W E ( R [ X ]) =0 , we have [ V ] = [ ψ d +12 ] . Thus upon applying ( [12, Lemma 5.5 and Lemma 5.6] ) , there exists γ ∈ SL d +1 ( R [ X ]) ∩ E d +2 ( R [ X ]) such that γ t V γ = ψ d +12 . By ( [2, Corollary 5.17] ) , e γ can be completed to an elementary matrix. Thusthere exists ε ∈ E d +2 ( R [ X ]) , and ρ ∈ SL d ( R [ X ]) ∩ E d +2 ( R [ X ]) such that ε t (1 ⊥ ρ ) t V (1 ⊥ ρ ) ε = ψ d +12 . By ( [1, Corollary 4.5] ) , there exists ε ∈ E d ( R [ X ]) such that (1 ⊥ ε ) t (1 ⊥ ρ ) t V (1 ⊥ ρ )(1 ⊥ ε ) = ψ d +12 . Now we set ρ = ( ρ ε ) − . Thus we have v = e ρ. (cid:3) Corollary 6.4.
Let R be a regular local ring of Krull dimension with ∈ R. Let v = ( v , v , v ) ∈ U m ( R [ X ]) . Then there exists ρ ∈ SL ( R [ X ]) ∩ E ( R [ X ]) such that v = e ρ . Proof : Since every v ∈ U m ( R [ X ]) is skew completable, thus the resultfollows in view of Lemma 6.3. (cid:3) OMPLETION OF SKEW COMPLETABLE UNIMODULAR ROWS 9 R EFERENCES [1] P. Chattopadhyay, R.A. Rao;
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AMPAT S HARMA , D
EAPRTMENT OF M ATHEMATICS , I
NDIAN I NSTITUTE OF T ECHNOLOGY B OMBAY , M
AIN G ATE R D , IIT A REA , P
OWAI , M
UMBAI , M
AHARASHTRA
E-mail : Sampat Sharma: Sampat Sharma