Adjamagbo Determinant and Serre conjecture for linear groups over Weyl algebras

Adjamagb o Determinan t and Serre Conjecture for linear groups o v er W eyl algebras Kossivi Adjamagb o Univ ersit ´ e P aris 6 - Case 172 - Institut de Mathmatiques de Jussieu 4, place Jussieu, 75 252 P ARIS CEDEX 0 5 adja@math.jussieu.fr Abstract: Tha nks to the th eory of dete rminants o ver an Or e domain, also called Adjamagb o determinan t b y the Russian school of non commuta tiv e algebra, we extend to any W eyl algebra ov er a field of characteristic zero Suslin theorem solving what Suslin him se lf called the K 1 -analogue of the well-kno wn Serre Conjecture and asserting that for any integer n gr e ater than 2, any n by n matrix with co effic ient s i n an y algebra of p olynomials ov er a field and with determinant one i s the pro duct of elemen tary matri ces wi th coefficients in this algebra . In t ro duction Let A be a ring, n a p ositiv e integer, M n ( A ) the ring of n b y n matrices with co eficien ts in A , GL n ( A ) the group of in v ertible elemen ts of M n ( A ) and E n ( A ) its subgroup generated b y its elemen tary matrices. Let us consider t he following natura l question : Do es there exist a function of the coefficien ts of the elemen ts of M n ( A ) with v alues somewhere whic h allo ws t he c hara cterization of elemen ts of E n ( A ) b etw een t hose of M n ( A ) and no t only b et w een those o f GL n ( A ) and suc h that this f unction is “effectiv ely computable” whenev er the ring A is “ effectiv e” in the sense o f [3]? Let us first a ssume A comm utativ e. In this case, w e think naturally to the de- terminan t function of matrices in M n ( A ). Indeed, fo r an y elemen t a of M n ( A ), if a ∈ E n ( A ), then it is obvious that det a = 1. But the con v erse is false in general, as sho wn b y the famo us counter-ex ample of P .M. Cohn [1 0], prop. 7.3 : a =  1 − X Y − Y 2 X 2 1 + X Y  ∈ M 2 ( Q [ X, Y ]) Nev ertheless, it is w ell kno wn that this con v erse is true if A is a field, or an euclidian ring or a semi-lo cal one, or if A is a no etherian ring whose Krull dimension is lo w er than n − 1, according to Bass-Milnor-Serre theorem on pro ducts of elemen ta ry matrices published in 1967 [9]. On the o ther hand, Suslin prov ed ten y ears later in [1 8] that t his last tho erem can b e impro ved by taking n only great er than 2 (instead of 1 plus the Krull dimension of A ) and b y c ho osing A a s any algebra of p olynomials o v er a field. In the intro duction of this pap er, Suslin himself presen ted this result a s the solution to “the K 1 -analogue of the w ell-known serre Problem recently solv ed completely by the author, and indep enden tly b y D aniel Quillen”. Indeed, a ccording to t he trivialit y of the sp ecial Withehead group S K 1 A o f suc h algebra, for a square matrix a with en tries in A , det a = 1 means that this matrix is “stably” (i.e. after aug mentation of the matrix b y adding some 1’s on the diagonal and 0 outside) a pro duct of elemen ta r y square matrices with en t r ies in A , and the problem is to kno w if it is “actually” a pro duct of elemen tary square matrices with en tries in A . This justifies Suslin’s analog y with 1 2 Serre problem whic h ask if any “stably” (i.e. after addition b y a finite free A -mo dule) free A-mo dule of finite t yp e is “ actually” free. Thanks to the theory of determinan ts ov er an Ore domain deve lopp ed in [4], summed up in [5], more brieve ly in [8], A,II I, and already called “Adjamagb o deter- minan t” b y Russian sc ho ol of non commutativ e alg ebra following A. Mikhalev and A. G uterman (see for instance [14], [15], [11], [1 2 ], [13]), we extended in [5], this theorem o f Bass-Milnor-Serre to the case where A is a non- comm utative “classical filtered ring”, i.e. a ring endo wed with a n increasing N - filtration F w hose asso ci- ated graded ring is a commutativ e regular domain flat o v er its subring F(0) whic h has a trivial sp ecial Whitehead group S K 1 F (0). It is in particular the case o f the en volopping a lgebra of a Lie algebra of finite dimension ov er a field. It is a lso the case of a classical or a fo r mal (resp. analytic) W eyl algebra o v er a field (resp. the field of real or complex num b ers), see for instance [5], p. 404. Exactly as Suslin did fo r Bass-Milnor-Serre theorem, the a im of this note is to impro ve this generalization of Bass-Milnor-Serre theorem, b y taking n o nly greater than 2 (instead o f 1 plus the Krull dimension of A ) and by c ho osing A as an y W eyl algebra o ver a field of c hara cteristic zero, tha nks to Stafford Main Theorem on the “ mo dule structure of W eyl a lg ebras” [1 7] and to V arerstein K 1 -stabilit y theorem [19], th. 3 .2 . Finally , we form ulate a natura l conjecture ab out filtered rings which w ould prov e that this Suslin Theorem follows from the forcomming theorem. W e end with other op en problems related to this theorem. Recall (on the canonical determinan t ov er filtered W eyl alg ebras, see fo r instance [5]) 1) K b eing a comm utativ e field and m a p ositiv e integer, let us denote by ( X i , Y i ) 1 ≤ i ≤ m a system of indeterminates ov er K , K [ X 1 , . . . , X m ] the K -algebra of p olynomials in indeterminates X 1 , . . . , X m , A m ( K ) the W eyl algebra ov er K of index m , i.e. the K -algebra K [ X 1 , . . . , X m ][ ∂ /∂ X 1 , . . . , ∂ /∂ X m ] o f K - linear differen tial op erators o v er K [ X 1 , . . . , X m ], F the differen tial filtratio n on A m ( K ), i.e. F (0) = K [ X 1 , . . . , X m ] and F ( j + 1) = F ( j ) + P 1 ≤ i ≤ m F ( j ) ∂ / ∂ X i for eac h natural in teger j , g r F A m K the associated graded ring, g r F the canonical ma p from A m ( K ) to g r F A m K and g r F ( A m ( K )) the image of g r F , i.e. the set of homogeneous elemen ts of the g raded ring g r F A m K , or in terms of partial differen tial equations, the set of principal sym- b ols of the differen tial op erato rs b elonging to A m ( K ). 2) The ring g r F A m ( K ) b eing isomorphic to the regular ring of p olynomials o v er K in indeterminates X 1 , . . . , X m , Y 1 , . . . , Y m and the group S K 1 F (0) = S K 1 K [ X 1 , . . . , X m ], isomorphic to S K 1 K a ccording to Quillen Theorem [1 6], 6, th. 7, b eing trivial, F is therefore a “classic regular filtration” on A m ( K ) making the later b e a “classic regular ring” and in particular an Ore domain. 3) According to the theory of determinan ts o v er Ore domains, there exists a n unique map from the set of square matrices o f elemen ts of A m ( K ) and with v alues in g r F ( A m ( K )), denoted b y det F and called “ the canonical determinant ov er the filtered W eyl alg ebra o ve r K of index m ” or “the principal determinan t ov er the 3 W eyl a lg ebra ov er K of index m ”, suc h that, for an y square matrices a , b of the same size with co efficien ts in A m ( K ) and an y diagonal matrix diag ( x, 1 , . . . , 1) with diagonal co efficien ts x, 1 , . . . , 1 in A m ( K ), we ha ve: det F ( ab ) = det F ( a )det F ( b ) ( the homomor phism axiom ) det F ( diag ( x, 1 , . . . , 1)) = g r F ( x ) ( the pr ol ong ation axiom ) 4) The first fondamental prop ert y following from this homomorphism axiom is that for an y elemen tary e with co efficien ts in A m ( K ), we hav e: det F ( e ) = 1 ( the elementar y pr oper ty ) 5) The second fo nda mental pro p ert y followin g easily fro m t hese t w o axioms and this elemen tary prop ert y is that for any triangular matrix t with co efficien t in A m ( K ), det F ( t ) is the pr o duct o f the principal sym b ols of its dia g onal co efficien ts (t he tri- angular prop erty) 6) The third fondaman tal prop ert y follo wing from these t w o axioms and this trian- gular prop erty is tha t for a n y n b y n matrix a with co efficien ts in A m ( K ), det F ( a ) can b e computed in a pratical w ay b y “G auss metho d”. Indeed, thanks to t he left comm un m ultiple prop erty of the Ore domain A m ( K ) and to suitable com binat io ns on the lines o f square matr ices with co efficien ts in A m ( K ), it is p ossible to find n b y n elemen tary or cancellable diagonal matrices p 1 , . . . , p r with co efficien ts in A m ( K ) suc h that p 1 . . . p r a is an upp er triangular matrix t . Then, thanks to the homomorphism axiom and the tr ia ngular prop erty , we o bta in the follo wing explicit expression of det F ( a ) as quotien t of tw o principal sym b ols of elemen ts of A m ( K ) whic h can finally b e reduced to the principal sym b ol of a elemen t o f A m ( K ) thanks to the factoriality of the ring g r F A m ( K ) and to the “regularity theorem” : det F ( a ) = Y 1 ≤ i ≤ n g r F ( t ( i, i )) / Y 1 ≤ k ≤ r Y 1 ≤ i ≤ n g r F ( p k ( i, i )) 7) If the field K is “effectiv e” in the sense o f [2], then it follows from this last form ula and from the “effectivit y” of t he Ore prop ert y of the “effectiv e ring” A m ( K ) that the restriction of det F to eac h “ effectiv e ring” of square matrices with co efficien ts in A m ( K ) is an “ effectiv ely computable” function, as pro v ed in [1]. 8) It also follows from this formula that if a a nd b are tw o square matrices with co efficien ts in A m ( K ) and if a ⊕ b =  a 0 0 b  then det F ( a ⊕ b ) = det F ( a )det F ( b ) 9) The link b et w een the classical determinan t o v er a comm utat ive ring and det F is that if B is an y comm utative sub-ring of A m ( K ) and det the classical determinan t o v er B , then : det F | B = g r F ◦ det 10) Finally , o ne of the most remark able analogies already disco v ered b et w een the classical determinant ov er a comm utativ e ring and det F is that an elemen t a of a ring of square matrices ov er A m ( K ) is in vertible if and only if det F ( a ) is inv ertible 4 in g r F A m ( K ), i. e. a non zero elemen t of K . Theorem F or an y integer n gr eater than 2, a n by n matrix a with co efficien ts in a n y W eyl algebra o ver an y field of c hara cteristic zero is a pro duct of elemen tary matrices with co efficien t in this a lgebra if and only its canonical determinan t ov er this a lg ebra is 1 . Pro of 1) Let A m ( K ) b e suc h a W eyl a lg ebra and F it s differen tial filtration. 2) If a is a pro duct of ele men tary matrices with co efficien ts in A m ( K ), then accord- ing to the homomorphism axiom and the elemen tary prop ert y of det F , it is ob vious that det F ( a ) = 1. 3) So let us assume conv ersly that det F ( a ) = 1. According the ab ov e c hara cteriza- tion of elemen ts o f GL n ( A m ( K )) b etw een those of M n ( A m ( K )), a ∈ GL n ( A m ( K )). 4) F b eing a classical regular filtra t ion as w e remark ed a b ov e, it follows from the cited Quillen theorem that the canonical map fr om K 1 F (0) in K 1 A m ( K ) is a n isomorphism. So there exists a po sitiv e in teger p , a p by p matrix b with co efficien ts in F (0) and in tegers r and s suc h that n + r = p + s and : ( ∗ ) ( a ⊕ i r ) − 1 ( b ⊕ i s ) ∈ E n + r ( A m ( K )) where i j denotes the unit of the group G L j ( A m ( K )) . 5) According to (*) and to the p oin ts 3), 4), 8) and 9) of the recall, w e hav e : det F ( b ⊕ i s ) = det( b ⊕ i s ) = 1 6) Since F is a classical regular filtr ation, in particular since S K 1 F (0) is trivial, it follo ws from this last equalit y that there exists a p ositif interger t suc h that : b ⊕ i s + t ∈ E p + s + t ( F (0)) 7) So according to (*), w e ha ve : a ⊕ i r + t ∈ E n + r + t ( A m ( K )) 8) On the other hand, according to Stafford cited theorem, the stable ra ng e of A m ( K ) is 2. Thanks to V aserstein cited theorem, it follo ws fro m the la st relation as desired that : a ∈ E n ( A m ( K )) Q.E.D. Remark 1 1) The previous theorem may by in terpreted in terms of systems o f partial differ- en tial equations with solutions in an y K [ X 1 , . . . , X m ]-algebra B whic h is a A m ( K )- left-mo dule, more precisely in terms of suc h a sy stem whic h is “eleme n ta ry resoluble b y deriv ation”, followin g and refining [1]. 5 2) The statemen t of the previous theorem is clearly the faithfull generalization to W eyl a lg ebras o v er a field of c haracteristic zero of Suslin theorem for algebras of p olynomials o ver suc h a field. 3) F urthermore, it solv es t he K 1 -analogue of Serre Conjecture ov er W eyl algebras, since according to p oints (2) a nd (7) of the previous pro of, for any square matrix a with en t ries in a W eyl algebra o v er a n y field o f c ha r a cteristic zero A , det F ( a ) = 1 means that this matrix is “stably” a pro duct of elemen tary square matrices with en tries in A . 4) On the other hand, Suslin theorem would follow fro m the prev ious theorem, with a non comm utativ e algebraic proo f completely different from Suslin or iginal com- m utat ive algebraic pro of, if the following “ natural” conjecture could b e confirmed : Conjecture If A is a ring endo wed with an increasing N - filtration F suc h tha t the asso ciated graded ring is a domain, then for any in teger n ≥ 2, the sub-ring F (0) of A v erifies : GL n ( F (0)) ∩ E n ( A ) = E n ( F (0)) Remark 2 1) The assumption on the asso ciated graded ring means that the asso ciated degree function deg F is ”additiv e”, i.e. is an homomorphism fro m ( A − { 0 } , × ) to ( N , +). Morev er, this degree f unction is suc h that F (0) − { 0 } = deg F − 1 (0). 2) Using t his degree fonction, it seems that the pro of of the conjecture could b e purely formal, as this statemen t could b e easily c hec k ed fo r n = 2 in the case where the considered elemen t of GL n ( F (0)) ∩ E n ( A ) is a pro duct of at most fiv e elemen ts of A . 3) The main in terest of this conjecture is that thanks to it, a non trivial prop ert y of a commutativ e ring (Suslin theorem) could be deduced from the similar pro p ert y of a “simple” non comm utativ e extension of this ring (previous theorem), in an analogous w a y as some dee p prop erties of the field o f real n um b ers could be deduced from the similar ones of a n “algebraically close d” extension of this field lik e the field of complex nu m b ers. 4) This kind of con tribution of non commutativ e algebra to comm utativ e algebra seems non common in mathematics. A confirmed example of suc h a con tribution is the fa ct t ha t the famous Jacobian Conjecture, whic h claim that an y endomorphism of an alg ebra o f p o lynomials o v er a field of characteristic zero with a non zero jaco- bian in this field is an automorphism, could b e deduced from Dixmier Conjecture, whic h claims that an y endomorphism of a W eyl algebra ov er a field of c haracteristic zero is an a utomorphism (see for instance [7], p. 2 97). 5) Another inte rest of the propo sed conjecture is that it w ould pro v e that Cohn coun ter- example cited in the in t r o duction is ev en b etter than one thinks now, in the sense that it is not in E 2 ( A 2 ( Q )), sho wing in this w ay that the low er b ound 3 for n in the previous theorem is the finest as in Suslin theorem. 6 6) Conformly to the “effectiv eness” problem ev o cat ed in the introduction a nd the recall, a natur a l question risen f r o m the previous theorem is the following : F or an in teger n greater than 2, ho w to split “ effectiv ely” a n by n matrix with co efficien ts in a W eyl algebra o v er an “effectiv e” field of c haracteristic zero and with principal determinant 1 as a pro duct of elemen tary o nes ? 7) According to the prominen t part that the cited Stafford stable rank theorem pla ys in the pro of of the previous theorem, it is easy to conjecture that this last “effectiv eness” problem should need the resolution of the following one : Giv en three elemen ts a, b, c of a W eyl algebra o ver an “effectiv e” field, ho w to find “effectiv ely” t w o eleme n ts d and e of this alg ebra suc h that a + dc and b + ec generate that same left ideal of this algebra as a , b and c ? 8) Let us now consider a last question risen from the previous t heorem. Indeed, f o r the problems met in the algebraic theory of partial differen tial equations, the working no etherian domains of differen tia l op erator a re not W eyl algebras, but what could b e called “formal (resp. conv ergen t)” W eyl algebras, deduced from “classical” W eyl algebras b y replacing p olynomials b y formal (resp. con vergen t) p ow er series (see for instance [5], p. 404). Since it is na tural to hop e that the previous theorem w orks also for these “w o rking” W eyl algebras, the pro o f of this theorem lead us to the following ques tion : Is the stable rank of a “formal” (resp. “con v ergen t”) W eyl a lgebra ov er a field of c hara cteristic zero (resp. a sub-field o f the field of complex n umbers) also 2 ? Reference s [1] K. Adjamagb o, Su r le gr oup e de Whitehe ad et les syst` emes d’´ equations aux d ´ eriv´ ees p artiel les , C.R.Acad.Sc. Paris, t. 299, serie I, n. 7, 1984, p. 205-2 08. [2] K. 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