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An Algorithmic Proof of Suslin's Stability Theorem Over Polynomial

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An Algorithmic Proof of Suslin's Stability Theorem Over Polynomial An Algorithmic Proof of Suslin’s Stability Theorem over Polynomial Rings Hyungju Park∗ Cynthia Woodburn† Abstract Let k be a field. Then Gaussian elimination over k and the Eu- clidean division algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SLn(k) or SLn(k[x]), n ≥ 2, as a product of elementary matrices. Suslin’s stability theorem states that the same is true for the multivariate polynomial ring SLn(k[x1,...,xm]) with n ≥ 3. As Gaussian elimination gives us an algorithmic way of finding an explicit factorization of the given matrix into elementary matrices over a field, we develop a similar algorithm over polynomial rings. 1 Introduction Immediately after proving the famous Serre’s Conjecture (the Quillen-Suslin theorem, nowadays) in 1976 [11], A. Suslin went on [12] to prove the following K -analogue of Serre’s Conjecture which is now known as Suslin’s stability arXiv:alg-geom/9405003v1 10 May 1994 1 theorem: Let R be a commutative Noetherian ring and n ≥ max(3, dim(R)+ 2). Then, any n × n matrix A =(fij) of determinant 1, with fij being elements of the polynomial ring R[x1,...,xm], can be writ- ten as a product of elementary matrices over R[x1,...,xm]. ∗Dept. of Mathematics, University of California, Berkeley; [email protected] †Dept. of Mathematics, Pittsburg State University; [email protected] 1 Definition 1 For any ring R, an n × n elementary matrix Eij(a) over R is a matrix of the form I + a · eij where i =6 j, a ∈ R and eij is the n × n matrix whose (i, j) component is 1 and all other components are zero. For a ring R, let SLn(R) be be the group of all the n × n matrices of deter- minant 1 whose entries are elements of R, and let En(R) be the subgroup of SLn(R) generated by the elementary matrices. Then Suslin’s stability theorem can be expressed as SLn(R[x1,...,xm]) = En(R[x1,...,xm]) for all n ≥ max(3, dim(R)+2). (1) In this paper, we develop an algorithmic proof of the above assertion over a field k. By implementing this algorithm, for a given A ∈ SLn(k[x1,...,xm]) with n ≥ 3, we are able to find those elementary matrices E1,...,Et ∈ En(k[x1,...,xm]) such that A = E1 ··· Et. Remark 1 If a matrix A can be written as a product of elementary matrices, we will say A is realizable. • In section 2, an algorithmic proof of the normality of En(k[x1,...,xm]) in SLn(k[x1,...,xm]) for n ≥ 3 is given, which will be used in the rest of paper. • In section 3, we develop an algorithm for the Quillen Induction Process, a standard way of reducing a given problem over a ring to an easier problem over a local ring. Using this Quillen Induction Algorithm, we reduce our realization problem over the polynomial ring R[X] to one over RM [X]’s, where R = k[x1,...,xm−1] and M is a maximal ideal of R. • In section 4, an algorithmic proof of the Elementary Column Property, a stronger version of the Unimodular Column Property, is given, and we note that this algorithm gives another constructive proof of the Quillen-Suslin theorem. Using the Elementary Column Property, we show that a realization algorithm for SLn(k[x1,...,xm]) is obtained 2 from a realization algorithm for the matrices of the following special form: p q 0 r s 0 ∈ SL3(k[x1,...,xm]), 0 0 1 where p is monic in the last variable xm. • In section 5, in view of the results in the preceding two sections, we note that a realization algorithm over k[x1,...,xm] can be obtained from a p q 0 realization algorithm for the matrices of the special form r s 0 0 0 1 over R[X], where R is now a local ring and p is monic in X. A realiza- tion algorithm for this case was already found by M.P. Murthy in [4]. We reproduce Murthy’s Algorithm in this section. • In section 6, we suggest using the Steinberg relations from algebraic K-theory to lower the number of elementary matrix factors in a factor- ization produced by our algorithm. We also mention an ongoing effort of using our algorithm in Signal Processing. 2 Normality of En(k[x1,...,xm]) in SLn(k[x1,...,xm]) 1+ xy x2 Lemma 1 The Cohn matrix A = 2 is not realizable, but −y 1 − xy ! A 0 ∈ SL3(k[x, y]) is. 0 1 ! Proof: The nonrealizability of A is proved in [1], and a complete algorithmic criterion for the realizability of matrices in SL2(k[x1,...,xm]) is developed in [13]. Now consider 1+ xy x2 0 A 0 = −y2 1 − xy 0 . (2) 0 1 ! 0 01 3 1+ xy x2 0 x 2 Noting that −y 1 − xy 0 = I + −y · (y, x, 0), we see that the 0 01 0 realizability of this matrix is a special case of the following Lemma 3. ✷ Definition 2 Let n ≥ 2. A Cohn-type matrix is a matrix of the form I + av · (vjei − viej) v1 . n where v = . ∈ (k[x1,...,xm]) , i < j ∈ {1,...,n}, a ∈ k[x1,...,xm], vn and ei = (0,...,0, 1, 0,..., 0) with 1 occurring only at the i-th position. Lemma 2 Any Cohn-type matrix for n ≥ 3 is realizable. Proof: First, let’s consider the case i =1, j = 2. In this case, v1 . B = I + a . · (v2, −v1, 0,..., 0) vn 2 1+ av1v2 −av1 0 ··· 0 2 av2 1 − av1v2 0 ··· 0 av v −av v = 3 2 3 1 . . . In−2 avnv2 −avnv1 2 1+ av1v2 −av1 0 ··· 0 2 av2 1 − av1v2 0 ··· 0 n 0 0 = El1(avlv2)El2(−avlv1),(3) . . l=3 . In−2 Y 0 0 So, it’s enough to show that 2 1+ av1v2 −av1 0 2 A = av2 1 − av1v2 0 (4) 0 01 4 is realizable for any a, v1, v2 ∈ k[x1,...,xm]. Let “→” indicate that we are applying elementary operations, and consider the following: 2 2 1+ av1v2 −av1 0 1+ av1v2 −av1 v1 2 2 A = av2 1 − av1v2 0 → av2 1 − av1v2 v2 0 01 0 01 2 1 −av1 v1 1 0 v1 1 0 v1 → 0 1 − av1v2 v2 → 0 1 v2 → 0 1 v2 −av 0 1 −av av 1 0 av 1+ av v 2 2 1 1 1 2 10 0 10 0 1 0 0 → 0 1 v2 → 0 1 v2 → 0 1 0 . (5) 0 av 1+ av v 00 1 0 0 1 1 1 2 Keeping track of all the elementary operations involved, we get A = E13(−v1)E23(−v2)E31(−av2)E32(av1)E13(v1)E23(v2)E31(av2)E32(−av1). (6) In general (i.e., for arbitrary i < j), v1 . B = I + a . · (0,..., 0, vj, 0,..., 0, −vi, 0,..., 0) vn (Here, vj occurs at the i-th position and −vi occurs at the j-th position.) 1 ··· av1vj ··· −av1vi ··· 0 . .. 0 2 1+ avivj −avi . = . 2 avj 1 − avivj . . v v −v v 1 n j n i 5 1 ··· 0 ··· 0 ··· 0 . .. 0 2 1+ avivj −avi . = . 2 avj 1 − avivj . . 0 0 1 · Eli(avlvj)Elj(−avlvi) 1≤l≤Yn,l=6 i,j = Eit(−vi)Ejt(−vj)Eti(−avj)Etj(avi)Eit(vi)Ejt(vj)Eti(avj)Etj (−avi) · Eli(avlvj)Elj(−avlvi). (7) 1≤l≤Yn,l=6 i,j In the above, t ∈ {1,...,n} can be chosen to be any number other than i and j. ✷ Since a Cohn-type matrix is realizable, any product of Cohn-type matrices is also realizable. This observation motivates the following generalization of the above lemma. t n Definition 3 Let R be a ring and v = (v1,...,vn) ∈ R for some n ∈ IN. Then v is called a unimodular column vector if its components generate R, i.e. if there exist g1,...,gn ∈ R such that v1g1 + ··· + vngn =1. Corollary 1 Suppose that A ∈ SLn(k[x1,...,xm]) with n ≥ 3 can be written in the form A = I +v·w for a unimodular column vector v and a row vector w over k[x1,...,xm] such that w · v =0. Then A is realizable. t Proof: Since v = (v1,...,vn) is unimodular, we can find g1,...,gn ∈ k[x1,...,xm] such that v1g1 + ··· + vngn = 1. We can use the effective Nullstellensatz to explicitly find these gi’s (See [3]). This combined with w · v = w1v1 + ··· + wnvn = 0 yields a new expression for w: w = aij(vjei − viej) (8) Xi<j 6 where aij = wigj − wjgi. Now, A = (I + v · aij(vjei − viej)) . (9) i<jY Each component on the right hand side of this equation is a Cohn-type matrix and thus realizable, so A is also realizable. ✷ −1 Corollary 2 BEij(a)B is realizable for any B ∈ GLn(k[x1,...,xm]) with n ≥ 3 and a ∈ k[x1,...,xm]. Proof: Note that i =6 j, and −1 −1 BEij(a)B = I +(i-th column vector of B) · a · (j-th row vector of B ). Let v be the i-th column vector of B and w be a times the j-th row vector of B−1. Then (i-th row vector of B−1) · v = 1 implies v is unimodular, and −1 w · v is clearly zero since i =6 j.
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