Unipotent Matrices, Modulo Elementary Matrices, in SL over a Coordinate Ring

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1 JOURNAL OF ALGEBRA 3, ARTICLE NO. JA9773 Unipotent Matrices, Modulo Elementary Matrices, in SL over a Coordinate Ring A. W. Mason* Department of Mathematics, Uniersity of Glasgow, Glasgow, G1 8QW, Scotl, United Kingdom Communicated by Leonard L. Scott, Jr. Received February 19, 1997 INTRODUCTION Let R be a Ž commutative. integral domain let R* be its set of units. A matrix U in GL R for which U I is called unipotent. Itis clear that the unipotent matrices U in GL Ž R. are precisely those for which det U 1 tr U. The simplest examples are the conjugates of the elementary matrices I re I re, where r R. Let U Ž R. 1 1 Žresp. NE Ž R.. be the subgroup Ž resp. normal subgroup. of GL Ž R. generated by the unipotent Ž resp. elementary. matrices. For each R-ideal, we put SL Ž R,. KerŽSL Ž R. SL Ž R.., the principal congruence subgroup of GL Ž R. of leel. Let U Ž R,. Žresp. NE Ž R,.. be the subgroup Ž resp. normal subgroup. of GL Ž R. generated by the unipotent Ž resp. elementary. matrices in SL Ž R,.. ŽBy definition, U Ž R, R. U Ž R. NE Ž R, R. NE Ž R... This paper is concerned with the structure of the quotient group U Ž R,. NE Ž R,.. This very much depends on R. Let P be a principal ideal domain let be any P-ideal. It is well known Žsee, for example, 9, Theorem that every unipotent matrix in SL Ž P,. is congruent to a matrix DE, modulo, where E is an elementary matrix in SL Ž P,. D is a diagonal matrix of determinant 1 in SL Ž P,.. Applying the so-called Whitehead Lemma 1, Ž 1.7. Proposition, p. 6 to D, it follows that U Ž P,. NE Ž P,.. Equalities of the type U Ž R,. NE Ž R,. can however hold Ž for non-zero. when R is not a principal ideal domain. Let A A be the S * awm@maths.gla.ac.uk $5. Copyright 1998 by Academic Press All rights of reproduction in any form reserved. 134

2 UNIPOTENT MODULO ELEMENTARY MATRICES 135 ring of S-integers in a global field F, where S is a finite, non-empty set of places of F, including all the archimedean places. Then A is a Dedekind domain often called an arithmetic Dedekind domain Žor a Dedekind ring of arithmetic type.. ŽSee 13, p Liehl 6 Vaserstein 17 have proved that, when A* isinfinite, the quotient group U Ž A,. NE Ž A,. is finite, cyclic. Moreover they proved Ž under the same hypothesis. that U Ž A. NE Ž A.Ž SL Ž A.. that, for most A, U Ž A,. NE Ž A,., where is any A-ideal. The simplest examples of A for which the latter equality holds are Ž in characteristic zero. the ring 1p, where is the ring of rational integers p is a rational prime, Ž in non-zero characteristic. 1 the Laurent polynomial ring kt, t, where k is a finite field. By a classical theorem of Dirichlet it follows that A* is finite if only if cardž S. 1. In characteristic zero therefore A* is finite if only if A or A O d, the ring of integers of the imaginary quadratic number field Ž ' d., where is the set of rational numbers d is a square-free, positive integer. Now is a Euclidean Ž hence principal ideal. domain so from the above U Ž,. NE Ž,., for all. It follows immediately from a result of Serre 13, Corollaire 3, p. 516 Žas shown by Grunewald, Mennicke, Vaserstein in the proof of 4, Proposition 5.5. that there exists an epimorphism : UŽ Od. NEŽ Od. n, where n 1 is the order of the ideal class group of O d. It follows from the above that U Ž O. NE Ž O. d d if only if Od is a principal ideal domain. ŽIt is a classical result due to Stark 15 that the only such d are d 1,, 3, 7, 11, 19, 43, 67, It is known 13, p. 489 that arithmetic Dedekind domains of non-zero characteristic with only finitely many units are of a single type. Such domains form a subfamily of the class of rings we are primarily concerned with in this paper. Let C be a smooth, projective curve over a field k, which is absolutely connected, let P be one of its closed points. By virtue of 16, I Corollary, p. 6 we may assume that k is algebraically closed in the function field, K, ofc. ŽIn this case k is said to be the full constant field 16, p. 1 of K.. Let C CŽ C, P, k. be the coordinate ring of the affine curve obtained by removing P from C. It is known 16, III., p. 67 that C is a Dedekind domain with quotient field K. The simplest example is C kt, the polynomial ring over k. It is also known that C* k*. When A has non-zero characteristic, it is known that A* is finite if only if A CŽ C, P, k., for some finite k. Our principal result is the following.

3 136 A. W. MASON THEOREM A. Suppose that C has a finite ideal class group. Then, for infinitely many C-ideals, there exist non-empty index sets, where, such that Ž. i UŽ C,. W, Ž ii. U Ž C,. NE Ž C,. W, where, for all, W V, the additie group of a countably infinite dimensional ector space V oer k. It is known 9, Sect. 3, for example, that the ideal class group of C is finite when k is finite or the genus of C is zero. In general some restriction on the C-ideals for which our theorem holds is necessary. We prove that a result like this does not hold for C when k has at least 4 elements. Our next result shows that a version of Theorem A does exist for C when k has at most 3 elements. THEOREM B. Suppose that cardž k. or 3. Let V be as aboe let m be the order of the ideal class group of C. Then there exist epimorphisms where m Ž. i :UŽ C. W, i i1 Ž ii. : U Ž C. NE Ž C. W, m i i Wi V Ž 1 i m.. From this a result of Grunewald, Mennicke, Vaserstein 4 we can prove the following COROLLARY C. Let ku, be the polynomial ring in u, oer k, where cardž k. or 3. Let V be as aboe. Then, for all positie integers n, there exists an epimorphism where Ž. Ž. : U k u, NE k u, W, i i1 W V 1 i n. i n

4 UNIPOTENT MODULO ELEMENTARY MATRICES 137 We conclude with a detailed study of the example C R x, y: x y 1, where is the set of real numbers. This example, first studied by Milnor 1, Example 13.5, p. 18, is of classical interest in algebraic K-theory. It is known that R is of the required type that the ideal class group of R has elements. ŽSee, for example, 9, Example In this paper we prove that U Ž R. NE Ž R.. This is in contrast with the corresponding situation for the class of rings Od described above. All of our results are based on a fundamental structure theorem for GL Ž C. due to Serre 14, Theorem 1, p We require a matrix formulation of Serre s theorem for which 9 is a convenient reference. We also require a number of stard results from the theory of algebraic function fields. A very useful reference for most of this material is the book of Stichtenoth 16. We adopt his notation as much as possible. As usual let denote the set of Ž strictly. positive integers. Large quotients arising from normal subgroups of GL Ž R. of the type described in this paper only occur when R has few units, for example, when R or C. When R has many units, quotients of this type are usually small. Moreover for such R it is often possible to classify the normal subgroups of GL Ž R. Ž or larger classes of subgroups. using special types of additive subgroups of R Ž for example, R-ideals.. Costa Keller 3 have done this for arithmetic Dedekind domains with infinitely many units. Menal Vaserstein 1, 11 have obtained similar classifications for most Ž not necessarily commutative. stable range one rings. 1. SERRE S THEOREM AND UNIPOTENT MATRICES Let C, P, K, k, C be as above. The closed points of C are in oneone correspondence with the places of the function field Kk. For each such point Q let OQ be the valuation ring of the corresponding discrete valuation. Then C can be defined in the following way C C Ž K, P, k. x K : x O Q, for all Q P 4. Ž. See 16, III.. Definition, p. 67. Let : K 4 be the discrete valuation of K corresponding to P. We record a number of basic properties of.

5 138 A. W. MASON Ž. LEMMA 1.1. a Let x C, where x. Ž. i Ž x., Ž ii. Ž x. if only if x k*. Ž iii. C* k*. Ž b. For each n, x C : Ž x. n4 is a finite-dimensional ector space oer k. Proof. Part Ž.Ž. a i follows from the product formula 16.I Theorem, p. 18. By the product formula Ž x. if only if x O, for all Q. Q By 16, I Corollary, p. 8 it follows that x lies in the algebraic closure of k in K. But k is assumed to be algebraically closed in K. Parts Ž.Ž. a ii Ž iii. follow. For part Ž b., see 16, I.4.9. Proposition, p. 18. As in 9, Sect. 1 we define a number of special subgroups of GL Ž C. each of which contains unipotent matrices. DEFINITION 1.. Let s K* n. For each k let 1 1 C, s C Cs Ž s Cs.. We define Ž. i 5 b Gn ½ :,k*, b C, Ž b. n, Ž ii. 5 b G G n ½ :, k*, b C, n1

6 UNIPOTENT MODULO ELEMENTARY MATRICES 139 Ž iii. sc Ž. s cs GnŽ s. ½ :, k*, c sc c C, s,ž c. n 5, Ž iv. sc Ž. s cs GŽ s. G n Ž s. ½ :, k*, c sc n1 c C, s. We now define subgroups of G Gs which consist entirely of unipotent matrices. DEFINITION 1.3. We define 1 b Ž. i U ½ : b C 1 5 ½ 1sc cs 5 1 c 1sc ii U s : c C Cs Cs C, s. Since C is a Dedekind ring its ideal class group can be identified 1, p. 14 with its Picard group, PicŽ C., there is a Ž natural. surjective map from K Ž the quotient field of C. to PicŽ C.. In this way we can use the elements of PicŽ C. to represent the conjugacy classes of unipotent matrices in GL Ž C.. LEMMA 1.4. Let the elements of Pic C be represented by s Ž., where s K*. Let U be a unipotent matrix in GL Ž C.. Then there exists g SL Ž C. such 1 1 that either gug G or gug Gs,for some unique. Proof. See 9, Theorem 1.11 To describe Serre s decomposition theorem for GL Ž C. we require some terminology from the theory of groups acting on trees. Let Ž G, Y. be a graph of groups 14, p. 37. This consists of a connected, oriented graph Y with vertex edge sets, vert Y edge Y, respectively, together with a rule which assigns groups to each vertex edge of 5

7 14 A. W. MASON Y. For each edge e of Y there corresponds another edge ež e. called the inerse of e. The groups corresponding to a vertex Ž resp. edge e. of Y are denoted by G Ž resp. G. e.if u, are the end-points of an edge e there are monomorphisms from Ge into Gu G. In addition we have GeG e, for all e edge Y. Associated with a graph of groups Ž G, Y. is its fundamental group Ž G,Y,T., whose definition involves a maximal subtree T of Y. See 1 14, p. 4. By 14, Proposition, p. 44 the definition of Ž G, Y, T. 1 is independent of the particular choice of T. In consequence the fundamental group is usually denoted by Ž G, Y. 1. As above we write the Picard group as where s K* Ž. We now define a tree T, where PicŽ C. 4 s : 4, Ž. a vert T, 4 : 4, Ž b. edge T e 4 e : 4, Ž. c the edge e Ž resp. e. joins Ž resp.., where. Let n n Ž. be positive integers let Ž H, T. be a graph of groups defined as Ž. i H GŽ., Ž ii. H GŽ s., Ž iii. H G Ž., Ž iv. H G Ž s.. e n e n Serre s theorem asserts that there exists a representation of PicŽ C. of this type, together with integers n n Ž. such that THEOREM 1.5 Serre. GL C H, T. 1 Proof. For the original version see 14, Proposition 1, p. 19. For this matrix formulation see 9, Theorem 1.1. Throughout most of this paper we will assume that PicŽ C. is finite. Serre s result however applies to any C.. THE PRINCIPAL RESULT In this section we restrict mainly to subgroups of proper congruence subgroups of GL Ž C.. Throughout this paper V denotes a countably infinite dimensional ector space oer k.

8 UNIPOTENT MODULO ELEMENTARY MATRICES 141 LEMMA.1. n, Let be a proper C-ideal. Then, for all s K* all 5 1 q q Ž. i UŽ C,. Gn ½ : q, Ž q. n, 1 Ž ii. UŽ C,. G ½ : q, 1sq qs 1 Ž iii. UŽ C,. GnŽ s. ½ : qs s, q 1sq Ž q. n 5, 1sq qs 5 1 q 1sq Ž iv. U Ž C,. GŽ s. : qs s. ½ Proof. Let X sc Ž. s cs c sc be an element of U Ž C,.. Then det X 1 tr X Ž mod.. Hence so 1. The rest follows. COROLLARY.. s K*, Let be a non-zero, proper C-ideal. Then, for all UŽ C,. G UŽ C,. GŽ s. V. Proof. By Lemma.1, UŽ C,. G UŽ C,. GŽ s. W, where W s 1 s. By 16, III..9. Proposition, p. 7 the maximal ideals of the Dedekind domain C are in oneone correspondence with the places of K other than P. From this correspondence it is clear from 16, I Proposition, p. 6 that dim kž C..

9 14 A. W. MASON Now W contains a non-zero C-ideal so it suffices to prove that C itself has countably infinite dimension over k. This follows from Lemma 1.1 together with Riemann s theorem 16, I Theorem, p. 1 DEFINITION.3. With the notation of Theorem 1.5, let S Ž resp. S. be a system of left coset representatives for GL Ž C. ŽU Ž C,..GŽ s.. Ž resp. GL Ž C. ŽU Ž C,..GŽ..., where. We define U g U Ž C,. GŽ s. g 1 Ž. gs Ž. U g U Ž C,. G g 1. gs We now come to our principal result. THEOREM.4. Suppose that PicŽ C. is finite. Let 4. Then, for infinitely many C-ideals, there exist S, where, such that Ž. i UŽ C,. U Ž ii. U Ž C,. NE Ž C,. U Ž.. Proof. We continue our use of the notation of Theorem 1.5. Let V c C : Ž c. n Ž.. 4 By Lemma 1.1Ž b., V is a finite-dimensional vector space over k so by 7, Lemma 3.1 there exist infinitely many non-zero ideals such that ½ 5 V 4. Let be one such ideal. By Lemma.1Ž. i, Ž iii. U Ž C,. G U Ž C,. G Ž s. I 4, n n for all. We now apply the subgroup theorem, Theorem 7, p. 11 for the fundamental group of a graph of groups to the Ž normal. subgroup U Ž C,. of GL Ž C.. By Theorem 1.5 there exist S, where, such that where H is a group. ž / U Ž C,. U Ž. H,

10 UNIPOTENT MODULO ELEMENTARY MATRICES 143 Let N be the normal subgroup of U Ž C,. generated by U Ž., where. Then by 9, Lemma.1 N is the normal subgroup of GL Ž C. generated by U Ž C,. X, where X G or Gs. Now, by definition, U Ž C,. is generated by unipotent matrices in SL Ž C,.. By Lemmas therefore UŽ C,. N so H 1. Part Ž. i follows. For part Ž ii. we note that by 9, Lemma.1 Lemma.1Ž ii. the normal subgroup of U Ž C,. generated by U Ž. is NE Ž C,.. COROLLARY.5. Suppose that Pic C is finite. Then, for infinitely many, there exist non-empty index sets, where, such that Ž. i UŽ C,. W, Ž ii. U Ž C,. NE Ž C,. W, where W V Ž.. Proof. This follows from Theorem.4 Corollary.. We note two important cases for which Pic C is finite. Ž. a The case where k is finite. When k is finite PicŽ C. is finite by 9, Theorem 3.3; 16, V.1.3, p Here each V is a finite set so the number of ideals for which ž / V 4 is finite. Theorem.4 applies in this case to all but finitely many ideals. Ž b. The genus zero case. Suppose that the genus 16, I Definition, p. 1 of K Ž or C. is zero. Then PicŽ C. is finite. ŽSee 9, Theorems 3.3, We consider in detail an important example of this type which we return to later on in the paper. Let K kt, Ž. the rational function field over any field k. Then k has genus zero by 16, I Example, p.. Let f be an irreducible polynomial of degree d Ž. over k. ŽWhen, for example, k is finite it is a classical result that irreducible polynomials of eery degree exist.. Now let g Af ½ : gk t, deg g nd, n 4. n f 5

11 144 A. W. MASON Then it is easily seen that A f C Ž C, P, k., where C is the projective line over k P is the point whose corresponding place in kt Ž. is determined by f. By 9, Theorem 3.4Ž ii. Pic A f d. For Corollary.5 to hold some restriction on is necessary. This is an immediate consequence of the following. LEMMA.6. Let G A, where A is abelian Ž.. Then no non-triial subgroup of G is perfect. Proof. Let S be a perfect subgroup of G. By the Kurosh subgroup theorem, ž / S S F, where F is a free group each S is a subgroup of a conjugate of some A. Let M ab denote the abelianization of an arbitrary group M. Then ž Ł / S ab F ab S. ab But S 1 so F S 1. Hence S 1. COROLLARY.7. Suppose that cardž k. 4. Then U Ž O. is not a free product of abelian groups. Proof. It is a classical result that SL Ž k. is generated by elementary matrices, since k is a field. It follows that SLŽ k. UŽ C.. Choose k* such that 1. Let x k. Then x 1 y 1 y, 1 1 1

12 UNIPOTENT MODULO ELEMENTARY MATRICES 145 Ž where y x It follows that SL Ž k. is perfect. We now apply Lemma.6. Corollary.7 contradicts an assertion made in the proof of 4, Proposition 5.6. The purpose of the next section is to show that a result similar to Corollary.5 does hold for the case C when cardž k. 3. DEFINITION VERY SMALL CONSTANT FIELDS For each s K* n we define 5 1 b Un ½ : b C, Ž b. n 1 1 sc cs 1 c 1sc 5 U Ž s. : c C Cs Cs,Ž c. n. n ½ Ž LEMMA 3.. Let s K* n. Then there exist epimorphisms of groups. such that such that : U V, Ž i. Un Ker, : UŽ s. V, Ž ii. UnŽ s. Ker. Proof. From the proof of Corollary., UŽ s. W, where W C Cs 1 Cs, a countably infinite dimensional k-vector space. By Lemma 1.1 b the subset 4 W x W : Ž x. n is a finite-dimensional subspace of W. Hence there exists an epimorphism of k-vector spaces : W V Ž. such that W Ker. The proof of part i is identical.

13 146 A. W. MASON Ž. THEOREM 3.3. Let card k or 3 let m card Pic C. Then there exist epimorphisms Ž. i : U Ž C. W, m i i1 Ž ii. : U Ž C. NE Ž C. W, where m i i Wi V Ž 1 i m.. Proof. ŽWe recall 9, Theorem 3.3 that PicŽ C. is finite when k is finite.. Our proof is based on Theorem 1.5. We represent PicŽ C. by, s 1,...,s m1, where s K* Ž 1im1. i. In the notation of Theorem 1.5 we take 1,...,m1. 4 Every value of k* is the determinant of a matrix in GŽ.. ŽSee, for example, 8, Corollary 3... This is also true for each Gs i, where 1 i m 1. Since k is finite we may assume that the integers n n Ž 1jm1. j are large enough to ensure that each of the subgroups G G Ž s. n n j, where 1 j m 1, also has this property. j Let N SL Ž C.. Then it follows from this assumption that N G N G Ž s. GL Ž C., n nj j where 1 j m 1. By Theorem 1.5, GL Ž C. is the fundamental group of a graph of groups, Ž H, T.. Applying, Theorem 7, p to the Ž normal. subgroup N of GL Ž C., it follows that where Ž as in Theorem 1.5. T vert T, N Ž L, T., 1 is a tree with 4,..., 4 1 m1 edge T e4 e 1,...,e m14. The edge e Ž resp. e. joins Ž resp.. j j, where 1 j m 1. The assignment L of groups to the vertices edges of T is as follows Ž. i L G N, Ž ii. L GŽ s. N, j j Ž iii. L G N, Ž iv. L G Ž s. N, e n ej nj j where 1 j m 1. We denote L by L.

14 UNIPOTENT MODULO ELEMENTARY MATRICES 147 We now apply the hypothesis card k 3. By the above we have Ž. a GNZU, Ž b. G NZU Ž. n n, Ž c. Gs NZUs j j, Ž d. G Ž s. NZU Ž s., nj j nj j where Z I 4 1 j m 1. Let W 1,...,Wm be m copies of V. By Lemma 3. there exist epimorphisms such that 1: G N W 1, G N Ker, n 1 : GŽ s. NW j j j1 such that G Ž s. NKer nj j j Ž 1jm1.. Since N Ž L, T. 1 we can extend these maps to an epimorphism m : N W i i1 in the natural way. Ž We define to be trivial on L.. Now G N UŽ C. GŽ sj. NUŽ C., where 1 j m 1. We now restrict to U Ž C. to get the required map. Part Ž. i follows. For part Ž ii. we replace above with the trivial map 1 1 : G N W 1. Extending this new set of maps as above restricting to U Ž C. we obtain the epimorphism. Theorem 3.3 has some implications for the structure of U Žku,., where ku, is the polynomial ring in variables u, over k.

15 148 A. W. MASON COROLLARY 3.4. Suppose that cardž k. or 3. For all n, there exists an epimorphism where : U Ž ku,. NE Ž ku,. W, n i i1 Wi V Ž 1 i n.. Proof. Let PŽ. be a polynomial in k of odd degree g 1, where g, which is not a square Žin kž.. let Aku,I, Ž. where I is the principal ideal generated by a u PŽ., when cardž k. 3, b u u PŽ., when cardž k.. Then A is an integral domain. Now let Q be the quotient field of A. Then Q FŽ y., where F kž x. y P x, when card k 3, or y y PŽ x., when cardž k.. It follows that Q: kž x. hence that Q is a hyperelliptic function field over k. ŽSee 16, VI., p By 16, VI..3, Proposition Ž. c, p. 194, together with the discussion on 16, p. 195, the pole of x Žregarded as a place of kž x.. extends to only one place, P*, say, of Q. It is easily seen that A is the intersection of the valuation rings of all the places of Q, apart from that of P*. From remarks made at the beginning it follows that A C Ž Q, P*, k.. Let J be the group of divisor classes of Q of degree zero. ŽSee 16, V.1.. Definition, p Now the place P* has degree one 16, I Definition Ž b., p. 6 so, using a stard short exact sequence 14, p. 14, it follows that PicŽ A. J. By 5, cardž J. is unbounded Ž as g increases.. ŽI am indebted to Professor R. W. K. Odoni for bringing this paper to my attention.. For all n therefore there exists A a epimorphism : U Ž A. NE Ž A. W, n i i1 by Theorem 3.3Ž ii.. The natural map : ku, Aextends to a homomorphism : GL Ž ku,. GL Ž A. this restricts to a homomorphism : ˆ U Ž ku,. U Ž A.. n n

16 UNIPOTENT MODULO ELEMENTARY MATRICES 149 ˆ Since A is a Dedekind domain, is surjective by 4, Lemma 5.3. The result follows. In 4, Proposition 5.6 Grunewald, Mennicke, Vaserstein assert that there is an epimorphism from U Žku,. onto a free product of an arbitrary finite number of groups, each of which is isomorphic to V, where k is any finite field. However, their proof contains the incorrect assertion that U Ž C. is such a free product, when C is in a hyperelliptic function field Ž over any finite field.. This has already been shown to be false in Corollary MILNOR S EXAMPLE In our final section we provide an example C for which Ž. i PicŽ C. is finite, non-trivial, Ž ii. U Ž C. NE Ž C.. This contrasts with the equivalent situation for the class of rings O d, where O is the ring of integers of the number field Ž ' d. Ž d with d.. As discussed in the Introduction it is known that U Ž O. d NE Ž O., when PicŽ O. d d is non-trivial. This example is a special case of a type introduced in Section. Let C be the projective line over, the set of real numbers. Here the function field is Ž. t which we denote by K. Let pž t. t 1, let : K 4 be the corresponding discrete valuation. Let P be the point of C corresponding to p Ž or.. We put R CŽ C, P,. g ½ : g t, deg g n, n 4. n p 5 Identifying x with tž1 t. y with Ž1 t. Ž1 t., it is clear that R x, y: x y 1. The ring R is of classical interest in algebraic K-theory since it is one of the first examples of a ring R shown Žby Milnor 1, Example 13.5, p. 18. to have non-trivial SK Ž R. 1. By 9, Corollary 3.5 we have PicŽ R.,

17 15 A. W. MASON we may represent the elements of Pic R Ž. See 9, Example 3.6. t. LEMMA 4.1. Let U GL Ž R. be unipotent. Then U is conjugate Ž oer SL Ž R.. to a matrix of one of the following types: 1 b Ž. i, where b R; 1 1ct ct 1 Ž ii., where c R Rt Rt. c 1ct Proof. This follows from Lemma 1.4. We note that here R Rt 1 Rt R Rt. Let O be the valuation ring of. Then we have by 5 f O ½ :f,gt,pg. g The group GL Ž K. Ž hence GL Ž R.. acts on X in the usual way, where X K. DEFINITION 4.. Let M be an O -module of rank contained in X. Then 4 StabŽ M. g GL Ž K.: gž M. M. We make use of the stard basis of X, which consists of e1 Ž 1,. e Ž,1.. DEFINITION 4.3. Let M be the rank, O -submodule of X spanned by te e pe. 1 1 Our proof that U Ž R. NE Ž R. is based on a number of properties of StabŽ M.. LEMMA 4.4. Let 1 g Ž SLŽ O.., 1 t a b g GLŽ R., c d

18 UNIPOTENT MODULO ELEMENTARY MATRICES a b gg g, c d where a d ct, b c,cbt da t c, d a ct. Then g StabŽ M. if only if Ž a., Ž b. 1, Ž c. 1, Ž d.. Proof. Let L be the O -submodule of X spanned by e pe. 1 1 Then g M L so g Stab M if only if g g g Stab L. LEMMA 4.5. Ž. i Stab M GL R, for all,, with. 1 1 t p tp Ž ii. StabŽ M. GL Ž R., where 1 1 tp t p, for all, *. Proof. This follows immediately from Lemma 4.4. We now come to less elementary properties. LEMMA 4.6. Ž. i StabŽ M. GL Ž R. GL Ž.. Ž ii. StabŽ M. SL Ž R. SL Ž., iii Stab M GL ½ :,,. Proof. defined by There is an epimorphism Ž of rings. 5 : R, Ž c. cž., this extends to a group homomorphism. ˆ : GL R Stab M GL.

19 15 A. W. MASON ˆ We show that is an isomorphism. We note that, from the definition, for all T. Ž. a is ˆ injectie. Let From Lemma 4.4, det ˆ Ž T. detž T., g a b Ker. ˆ c d Ž c. 1, 1 where c btdat c. Now cp R O so, by Lemma 1.1, b tž da. t cp, ˆ for some. Since g I, it follows that hence by Lemma 4.4 that 1 d ct gg g. act The elements a d ct d a ct are the eigenvalues of g so a, d R Ž since R is a Dedekind domain.. But ad det g *. Thus a, d*, since R* * by Lemma 1.1Ž iii.. Thus ct Ž. t ct g. c ct Note that det g. Again ˆŽ g. I By Lemma 4.4, Ž c. 1 so so 1 cž.. t t c, p for some,. If c, then ct R. It follows that c hence that ˆ 4 Ker I.

20 UNIPOTENT MODULO ELEMENTARY MATRICES 153 Ž b. ˆ is surjectie. Let H ImŽ ˆ.. By Lemma 4.5, H contains all the matrices, where,,,, *. It follows that H contains a matrix of the type 1 y, 1 for some y *. ŽNote that H contains, for example, / 1 x Let x *, where x y, let 1. Then y 1 y 1 1 y 1 x belongs to H. It is well known that GL is generated by matrices of the type 1 x 1,,, where x *. Part Ž. i follows. Part Ž ii. follows from Ž i. since ˆ preserves determinants. Part Ž iii. follows from Lemma 4.5Ž. i. THEOREM 4.7. Ž. i PicŽ R., Ž ii. NE Ž R. U Ž R.. Proof. By Lemma 4.1 it suffices to prove that UŽ c. 1 ct ct NEŽ R., c 1ct for all c R Rt. Let N NE Ž R.. Then N contains all conjugates of the matrices 1 r 1,

21 154 A. W. MASON where r R hence SL. In particular, N, for all,, where 1. By Lemma 4.6Ž iii. StabŽ M. N I 4. Since PSL is simple it follows from Lemma 4.6 ii that In particular StabŽ M. SLŽ R. N. dt d W Ž. N, 1 d dt Ž 1. 1 for all *, where d t p. We note the identities 1 c d gu c g gw g. 1 Let c R Rt let c c3. Using these identities it is easily verified that Ž Ž. W U c W U c U c. It follows from the above that Uc N. REFERENCES 1. H. Bass, Algebraic K-Theory, Benjamin, New York, D. E. Cohen Combinatorial Group Theory: A Topological Approach, London Mathematical Society Student Texts, Vol. 14, Cambridge Univ. Press, Cambridge, D. L. Costa G. E. Keller, The EŽ, A. sections of SLŽ, A., Ann. of Math. 134 Ž 1991., F. G. Grunewald, J. L. Mennicke, L. N. Vaserstein, On the groups SL Žx. SL Žkx, y., Israel J. Math. 86 Ž 1994., E. Inaba, Number of divisor classes in algebraic function fields, Proc. Japan Acad. 6, No. 7 Ž 195., B. Liehl, On the group SL over orders of arithmetic type, J. Reine Angew. Math. 33 Ž 1981., A. W. Mason, Free quotients of congruence subgroups of SL over a coordinate ring, Math. Z. 198 Ž 1988., A. W. Mason, Groups generated by elements with rational fixed points, Proc. Edinburgh Math. Soc. 4 Ž 1997., 193.

22 UNIPOTENT MODULO ELEMENTARY MATRICES A. W. Mason, Free quotients of congruence subgroups of SL over a coordinate ring unipotent matrices, submitted for publication. 1. P. Menal L. N. Vaserstein, On subgroups of GL over non-commutative local rings which are normalized by elementary matrices, Math. Ann. 85 Ž 1989., P. Menal L. N. Vaserstein, On the structure of GL over stable range one rings, J. Pure Appl. Algebra 64 Ž 199., J. Milnor, Introduction to Algebraic K-Theory, Annals of Mathematics Studies, Vol. 7, Princeton Univ. Press, Princeton, NJ, J.-P. Serre, Le probleme ` des groupes de congruence pour SL, Ann. of Math. 9 Ž 197., J.-P. Serre, Trees, Springer-Verlag, New YorkBerlin, H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 Ž 1967., H. Stichtenoth, Algebraic Function Fields Codes, Springer-Verlag, New YorkBerlin, L. N. Vaserstein, On the group SL over Dedekind rings of arithmetic type, Math. USSR-Sb. 18 Ž 197., 3133.