A note on cancellation of projective modules

A note on cancellation of pro jectiv e mo dules Alp esh M. Dho ra jia and Mano j K. Keshari Department of Mathematics, I IT Mumbai, Mumbai - 400076, Ind ia (alpesh,keshari)@math.iitb.ac.in 1 In tro duction Let A b e a ring o f dimension d and let P be a pr o jective A -module of rank d . Assume that if R is a finite extensio n of A then R d is cancellative. Then it is prov ed in ([10], Theorem 3.6 ) that P is also cancellative. In other words, if GL d +1 ( R ) acts tra nsitiv ely on Um d +1 ( R ) for every finite extension R of A , then Aut( A ⊕ P ) ac ts tra nsitiv ely o n Um( A ⊕ P ). W e will generalize the ab o ve re s ult as follows (3.4). Theorem 1.1 L et A b e a ring of dimension d and let P b e a pr oje ctive A -mo dule of r ank d . Assume that if R is a finite extension of A t he n E d +1 ( R ) acts tr ansitively on Um d +1 ( R ) . Then E ( A ⊕ P ) acts tr ansitively on Um ( A ⊕ P ) . If A is an affine algebra of dimension d ov e r Z then V aser stein [14] proved that E d +1 ( A ) acts transitively on Um d +1 ( A ). As a consequence o f (1.1) we get another pro of of the following re sult of Mohan Kumar, Murthy and Ro y ([11], Theor em 2.4) that if P is a pro jective A -mo dule of rank d , then E ( A ⊕ P ) acts transitively on Um( A ⊕ P ). Let A be a smo oth affine algebra of dimension d ov e r a n alge br aically closed field k . Assume that if characteristic of k is p > 0, then p ≥ d . Recently F a sel, Rao a nd Swan ([6], T heo rem 7.3) proved that stably fre e A - mo dules of rank d − 1 ar e free, thus ans wering an old ques tion of Sus lin. Infact if d ≥ 4, then they proved that A b eing normal s uffices. In view of their result, a natural question ar ises: Let P be a pro jectiv e A -mo dule of ra nk d − 1. Is P cancella tiv e? W e answer this que s tion in affirmative when k = F p . More precisely , we prove the following r esult (3 .5 ). Theorem 1.2 L et A b e an affine algebr a of dimension d ≥ 4 over F p , wher e p ≥ d . Then every pr oje ctive A -mo dule of r ank d − 1 is c anc el lative. Finally , w e will prov e the following result (4.5). Gubela dze pr o ved this result ([8], [9]) in case P is free. Theorem 1.3 L et M ⊂ Q r + b e a seminormal monoi d such that M ⊂ Q r + is an inte gr al extension. L et R b e a ring of dimension d and let P b e a pr oje ctive R [ M ] -mo dule of r ank n . Then E ( R [ M ] ⊕ P ) acts tr ansitively on Um ( R [ M ] ⊕ P ) whenever n ≥ max (2 , d + 1) . 2 Preliminaries All the rings are assumed to b e co mm utative No etherian and all the mo dules are finitely genera ted. Let A b e a r ing and let M be an A -mo dule. W e say that m ∈ M is unimo dular if ther e ex is ts φ ∈ M ∗ such that φ ( m ) = 1. The set of all unimo dular e lemen ts of M will b e denoted by Um( M ). W e denote by Aut A ( M ), the group o f a ll A -automorphism of M . F or an idea l J of A , we denote by Aut A ( M , J ), the kernel o f the natural homomorphism Aut A ( M ) → Aut A ( M /J M ). 1 W e denote b y E L 1 ( A ⊕ M , J ), the subg roup of Aut A ( A ⊕ M ) g enerated by all the automorphis ms ∆ aϕ =  1 aϕ 0 id M  and Γ m =  1 0 m id M  with a ∈ J , ϕ ∈ M ∗ and m ∈ M . W e will write E L 1 ( A ⊕ M ) for E L 1 ( A ⊕ M , A ). W e denote by Um 1 ( A ⊕ M , J ), the set of all ( a, m ) ∈ Um( A ⊕ M ) such that a ∈ 1 + J and by Um( A ⊕ M , J ), the se t of a ll ( a, m ) ∈ Um 1 ( A ⊕ M , J ) with m ∈ J M . W e will write Um 1 r ( A, J ) for Um 1 ( A ⊕ A r − 1 , J ) a nd Um r ( A, J ) for Um( A ⊕ A r − 1 , J ). Let M be an A -module. Let p ∈ M and ϕ ∈ M ∗ be such that ϕ ( m ) = 0. Let ϕ p ∈ E nd ( M ) be defined as ϕ p ( q ) = ϕ ( q ) p . Then 1 + ϕ p is a unip oten t automorphism o f M . An automo rphism o f M of the form 1 + ϕ p is called a tr ansve ction of M if either p ∈ Um ( M ) or ϕ ∈ Um( M ∗ ). W e denote by E ( M ), the subg roup of Aut( M ) generated by all the trans v ections of M . The following result is due to Ba k, B asu and Rao ([1], Theorem 3.10). In [5], w e hav e prov ed res ults for E L ( A ⊕ P ). Due to this result, w e can us e E ( A ⊕ P ) ev erywhere. Theorem 2.1 L et A b e a ring and let P b e a pr oje ctive A -mo dule of r ank ≥ 2 . Then E L ( A ⊕ P ) = E ( A ⊕ P ) . Remark 2.2 Using (2.1 ), it is easy to see that if I is any ideal of A , then the natural map E ( A ⊕ P ) → E (( A ⊕ P ) /I ( A ⊕ P )) is surjective. The following is a classica l r e sult due to Ba ss [2]. Theorem 2.3 L et A b e a ring and let P b e a pr oje ctive A -mo dule of r ank > dim A . Then E ( A ⊕ P ) acts tr ansitively on Um ( A ⊕ P ) . The following result is due to Lindel ([12], Lemma 1.1). Lemma 2.4 L et A b e a ring and let P b e a pr oje ctive A -mo dule of r ank r . Then ther e ex ists s ∈ A such that the fol lowing holds: ( i ) P s is fr e e, ( ii ) ther e ex ist s p 1 , . . . , p r ∈ P and φ 1 , . . . , φ r ∈ Hom( P , A ) such that ( φ i ( p j )) = diagonal ( s, . . . , s ) , ( iii ) sP ⊂ p 1 A + . . . + p r A , ( iv ) the image of s in A r ed is a non-zer o-divisor and ( v ) (0 : sA ) = (0 : s 2 A ) . The following tw o results ar e from ([5], Lemma 3.1 a nd Lemma 3.1 0). Lemma 2.5 L et A b e a ring and let P b e a pr oje ctive A -mo dule. L et “b ar” denote re duction mo dulo the nil ra dic al of A . If E ( A ⊕ P ) acts tr ansitively on Um( A ⊕ P ) , then E ( A ⊕ P ) acts tr ansitively on Um ( A ⊕ P ) . Lemma 2.6 L et A b e a ring and let P b e a pr oje ctive A -m o dule of r ank r . Cho ose s ∈ A , p 1 , . . . , p r ∈ P and ϕ 1 , . . . , ϕ r ∈ P ∗ satisfying the pr op erties of (2.4). L et ( a, p ) ∈ Um( A ⊕ P , sA ) with p = c 1 p 1 + . . . + c r p r , wher e c i ∈ sA for i = 1 , . . . , r . Assume ther e exists φ ∈ E 1 r +1 ( R, sA ) such that φ ( a, c 1 , . . . , c r ) = (1 , 0 , . . . , 0) . Then ther e exists Φ ∈ E ( A ⊕ P ) such that Φ( a, p ) = (1 , 0) . The following result is due to Mo han K umar, Murthy a nd Roy ([11], Theorem 2.4). Theorem 2.7 L et A b e an affine algebr a of dimension d ≥ 2 over F p . L et P b e a pr oje ctive A -mo dule of r ank d . Then E ( A ⊕ P ) acts tr ansitively on Um( A ⊕ P ) . The following result ([10], Theor em 3.8 ) is very crucial fo r the pro of o f (3.5 ). 2 Theorem 2.8 L et A b e an affine algebr a of dimension d over F p . Assume that if R is a finite extension of A t hen R d − 1 is c anc el lative. Then every pr oje ctive A -mo dule of r ank d − 1 is c anc el lative. W e end this se ction with a result due to F a sel, Rao and Swan ([6], Corollary 7.4). Prop osition 2 .9 L et R b e an affine algebr a of dimension d ≥ 4 over an algebr aic al ly close d field k . Assume that if char acteristic of k is p > 0 , then p ≥ d . L et J b e the ide al defining the singular lo cus of R . Then GL d ( R ) acts tr ansitively on Um d ( R, J ) . 3 Main Theorem In this section, w e prove our ma in re s ult. Let A b e a ring and I an ideal o f A . F or an integer n ≥ 3, define E n ( I ) a s the subgro up of E n ( A ) generated b y E ij ( a ) = I d + ae ij , where a ∈ I , 1 ≤ i 6 = j ≤ n and o nly non-ze ro entry of the matrix ae ij is a at the ( i, j )th pla ce. Consider the cartesian sq uare A ( I ) p 1 / / p 2   A j 1   A j 2 / / A/I The relative gro up E n ( A, I ) is defined in [13] by the exact se quence 1 → E n ( A, I ) → E n ( A ( I )) E n ( p 1 ) − − − → E n ( A ) → 1 and it is shown ([13 ], Pro position 2.2) that E n ( A, I ) is isomorphic to the kernel of the natural map E n ( A ) → E n ( A/I ). F ur ther , E n ( A, I ) is the normal closure of E n ( I ) in E n ( A ) [3]. Lemma 3.1 L et R b e a ring and I an ide al of R . If n ≥ 3 , then E n ( R, I 2 ) ⊂ E n ( I ) . Pro of It is enough to show that if β = E ij ( z ) ∈ E n ( I 2 ) and α = E kl ( z ′ ) ∈ E n ( R ) then αβ α − 1 ∈ E n ( I ). Note tha t e ij e kl = e il if j = k a nd 0 if j 6 = k . First ass ume that i 6 = l and hence ( i, j ) 6 = ( l, k ). Then αβ α − 1 = ( I d + z ′ e kl )( I d + z e ij )( I d − z ′ e kl ) = ( I d + z ′ e kl + z e ij )( I d − z ′ e kl ) = I d + z ′ e kl + z e ij − z ′ e kl − z ′ 2 e kl e kl − z ′ z e ij e kl = I d + z e ij − z ′ z e ij e kl . If j = k , then αβ α − 1 = I d + z e ij − z ′ z e il = ( I d + z e ij )( I d − z ′ z e il ) ∈ E ( I ). F urther, if j 6 = k , then αβ α − 1 = I + z e ij ∈ E ( I ). This proves that if i 6 = l then αβ α − 1 ∈ E ( I ). Similarly we can prov e that if j 6 = k then αβ α − 1 ∈ E ( I ). Now as sume that ( i, j ) = ( l , k ). Cho ose r ≤ n differ e n t from i, j and write z = a 1 b 1 + a 2 b 2 + . . . + a s b s , where a i , b i ∈ I . Now we ca n wr ite β = E ij ( z ) = s Y t =1 E ij ( a t b t ) = s Y t =1 [ E ir ( a t ) , E r j ( b t )] and αβ α − 1 = s Y t =1 [ αE ir ( a t ) α − 1 , αE r j ( b t ) α − 1 ] ∈ E n ( I )  Lemma 3.2 L et R b e a ring and I an ide al of R . If n ≥ 3 , then E n ( I ) ⊂ E 1 n ( R, I ) and henc e E n ( R, I 2 ) ⊂ E 1 n ( R, I ) . 3 Pro of Let E ij ( x ) ∈ E n ( I ), wher e x ∈ I . If i = 1 or j = 1, then E ij ( x ) ∈ E 1 n ( R, I ). Ass ume i 6 = 1 and j 6 = 1. Then E ij ( x ) = E i 1 (1) E 1 j ( x ) E i 1 ( − 1) E 1 j ( − x ) ∈ E 1 n ( R, I ).  Lemma 3.3 L et A b e a ring and let P b e a pr oje ctive A -m o dule of r ank r . Cho ose s ∈ A satisfying the c onditions in (2.4). Assu me t hat if R = A [ X ] / ( X 2 − s 2 X ) then E r +1 ( R ) acts tr ansitively on Um r +1 ( R ) . Then E ( A ⊕ P ) acts t ra nsitively on Um( A ⊕ P, s 2 A ) . Pro of Without lo ss of genera lit y , w e may a ssume that A is r e duced. By (2.4), there ex is t p 1 , . . . , p r ∈ P and φ 1 , . . . , φ r ∈ Hom( P , A ) such that P s is free, ( φ i ( p j )) = dia gonal ( s, . . . , s ), sP ⊂ p 1 A + . . . + p r A and s is a non-zer odiviso r . Let ( a, p ) ∈ Um( A ⊕ P , s 2 A ). Replacing p by p − ap , we may a s sume that p ∈ s 3 P . Since sP ⊂ P r 1 Ap i , we get p = f 1 p 1 + . . . + f r p r for so me f i ∈ s 2 A . Note that v = ( a, f 1 , . . . , f r ) ∈ Um r +1 ( A, s 2 A ). Consider the following cartesian sq ua re R p 1 / / p 2   A j 1   A j 2 / / A/ ( s 2 ) Patc hing unimo dular ro ws ( a, f 1 , . . . , f r ) and (1 , 0 , . . . , 0 ) ov er A/s 2 A , we get a unimodula r row ( c 0 , c 1 , . . . , c r ) ∈ Um r +1 ( R ). Since E r +1 ( R ) acts transitively on Um r +1 ( R ), there exists Θ ∈ E r +1 ( R ) such that ( c 0 , c 1 , . . . , c r )Θ = (1 , 0 , . . . , 0). The pro jections of this equation g iv es ( f , f 1 , . . . , f r )Ψ = (1 , 0 , . . . , 0) and (1 , 0 , . . . , 0) e Ψ = (1 , 0 , . . . , 0) where Ψ , e Ψ ∈ E r +1 ( A ) such that Ψ = e Ψ mo dulo ( s 2 ). Hence ( f , f 1 , . . . , f r )Ψ e Ψ − 1 = (1 , 0 , . . . , 0), where Ψ e Ψ − 1 = ∆ ∈ E r +1 ( A, s 2 A ). By (3.2), ∆ ∈ E 1 r +1 ( A, sA ). Hence b y (2.6), ther e exists Θ ∈ E ( A ⊕ P ) such that ( a, p )Θ = (1 , 0). This c ompletes the pro of.  Theorem 3.4 L et A b e a ring of dimension d and let P b e a pr oje ctive A -mo dule of r ank d . Assume that if R is a finite extension of A t he n E d +1 ( R ) acts tr ansitively on Um d +1 ( R ) . Then E ( A ⊕ P ) acts tr ansitively on Um ( A ⊕ P ) . Pro of Let ( a, p ) ∈ Um( A ⊕ P ). Cho ose s ∈ A satisfying the conditions in (2.4). Let “bar ” denote reduction mo dulo s 2 A . Since dim A = d − 1, by (2.3), there exists σ ∈ E ( A ⊕ P ) such that ( a, p ) σ = (1 , 0). By (2.2), we can lift σ to θ ∈ E ( A ⊕ P ). If ( a, p ) θ = ( b, q ), then ( b, q ) ∈ Um( A ⊕ P , s 2 A ). By (3.3 ), there exists θ 1 ∈ E ( A ⊕ P ) such that ( b, q ) θ 1 = ( a, p ) θ θ 1 = (1 , 0). This prov es the r esult.  Theorem 3.5 L et A b e an affine algebr a of dimension d ≥ 4 over the field F p , wher e p ≥ d . L et P b e a pr oje ctive A -mo dule of r ank d − 1 . Then P is c anc el lative. Pro of By (2.8), it is enough to show that if R is any a ffine a lgebra o f dimension d ov er F p , then R d − 1 is cancellative. Le t v ∈ Um d ( R ) be any unimo dular r o w of length d . It is enough to show that there exists σ ∈ GL d ( R ) such that v σ = e 1 = (1 , 0 , . . . , 0). Without los s of g eneralit y , we may assume that R is reduced. Let J be the idea l of R defining the singula r lo cus of R . Since R is reduced, height of J is ≥ 1. Let “bar ” denote re duction modulo J . Then dim R = d − 1 . By (2.7 ), there exits σ ∈ E d ( R ) such that 4 v σ = e 1 . By (2.2), we ca n lift σ to θ ∈ E d ( R ). W e hav e v θ = e 1 mo dulo J . Applying (2.9), we g et θ ∈ GL d ( R ) such tha t v σ θ = e 1 . Hence v is completable to an inv ertible matrix, i.e. R d − 1 is cancellative. This c ompletes the pro of.  4 Extension of Gub eladze’ s results In this section we extend some r esults of Gubeladze. W e b egin by r ecalling three results due to Gub eladze [7], ([8], Theorem 8 .1) a nd ([9 ], Theorem 10.1 ) resp ectiv e ly . See [9] for the definition o f a monoid M of Φ-simplicial g ro wth. Theorem 4.1 L et M b e a c ommut ative t orsio n-fr e e seminormal and c anc el lative monoid. Then for any princip al ide al domain R , pr oje ct ive mo dules over R [ M ] ar e fr e e. Theorem 4.2 L et R b e a ring of di mension d and let M ⊂ Q r + b e a submonoid such that M ⊂ Q r + is an inte gr al extension. Then E n ( R [ M ]) acts tr ansitively on Um n ( R [ M ]) whenever n ≥ max (3 , d + 2 ) . Theorem 4.3 L et R b e a ring of dimension d and let M b e a monoid of Φ -simplicial gr owth. Then E n ( R [ M ]) acts tr ansitively on Um n ( R [ M ]) whenever n ≥ max (3 , d + 2) . W e will generalize a bov e results a s follows. Theorem 4.4 L et M b e as in (4.2) or ( 4.3). L et R b e a ring of dimension d and let P b e a pr oje ctive R [ M ] -mo dule of r ank n . Assu m e that S − 1 P is fr e e, wher e S is the set of non-zer o divisors of R . Then E ( R [ M ] ⊕ P ) acts tr ansitively on Um ( R [ M ] ⊕ P ) whenever n ≥ max (2 , d + 1) . Pro of By (2.5), we may assume that the ring A = R [ M ] is reduced. W e will use induction on d . If d = 0, then by a ssumption, pro jectiv e mo dules o f consta n t rank over R [ M ] are free. Hence w e ar e done by (4.2) and (4 .3) . Assume d > 0. By assumption S − 1 P is free. W e ca n ch o ose s ∈ S s uch that P s is free and co nditions of (2 .4) a re sa tis fie d. Let ( a, p ) ∈ Um( A ⊕ P ) a nd let “bar ” deno te reduction modulo s 2 A . Since dim R = d − 1, by induction hypothesis, there exis ts φ ∈ E ( A ⊕ P ) such that ( a, p ) φ = (1 , 0). Let Φ ∈ E ( A ⊕ P ) b e a lift of φ , by (2.2). Then ( a, p )Φ ∈ Um ( A ⊕ P, s 2 A ). By Gubeladze’s theor em in the free case, E n ( B [ M ]) acts transitively on Um n ( B [ M ]), where B = R [ X ] / ( X 2 − s 2 X ). Hence b y (3.3), there exists Φ 1 ∈ E ( A ⊕ P ) suc h that ( a, p )ΦΦ 1 = (1 , 0). This c ompletes the pro of.  Using (4.1, 4.4), we get the following. Theorem 4.5 L et M b e as in (4.2) or (4.3). F urther assu me that M is seminormal. L et R b e a ring of dimension d and let P b e a pr oje ctive R [ M ] -mo dule of r ank n . Then E ( R [ M ] ⊕ P ) acts tr ansitively on Um( R [ M ] ⊕ P ) whenever n ≥ max (2 , d + 1) . Ac knowledgemen t: The first author would lik e to thank Profess or Nikolai V avilo v for useful discussion and second author w ould like to thank Professor Bha t wadek a r for pointing o ut a gap in an earlier version. 5 References [1] A. Bak, R. 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