HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS

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1 HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS XUEJUN GUO 1 ADEREMI KUKU 2 1 Department of Mathematics, Nanjing University Nanjing, Jiangsu , The People s Republic of China guoxj@nju.edu.cn The Abdus Salam International Center for Theoretical Physics, Trieste, Italy 2 Institute for Advanced Study, Princeton, NJ, USA kuku@ias.edu Abstract: In this paper we study the possible torsion in even dimensional higher class groups Cl 2n (Λ) (n 1) of an order Λ in a semi-simple algebra A over a number field F with ring of integers O F. We show that for certain orders called generalized Eichler orders p-torsion in Cl 2n (Λ) can only occurs for primes p dividing prime ideals of O F, at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras to hereditary orders. Keywords: higher class group, quaternion algebra, Eichler order, semi-simple algebra, hereditary order 2000 Mathematics Subject Classification: 19D50, 19F Introduction Let F be a number field O F the ring of integers in F. Let A be a semi-simple algebra over F Λ an O F -order in A. The higher class groups of Λ are defined as Cl n (Λ) = ker(sk n (Λ) SK n (Λ )), where runs through all maximal ideals of O F SK n (Λ) := ker(k n (Λ) K n (A)), for all integers n 0. By the Theorem 1 2 in [2], Cl n (Λ) are trivial for maximal orders. Later Kuku proved in [6] that Cl n (Λ) are finite for arbitrary orders. In [4], it is proved that the only p-torsion possible in Cl 2n+1 (Λ) is for those rational primes p which lie under the prime ideals of O F at which Λ is not maximal. In this paper, we prove the analogous result for even dimensional higher class groups Cl 2n (Λ) (n 1) in 1

2 the case where Λ is an Eichler order in a quaternion algebra or a hereditary order in a semi-simple algebra. Note that in general, an Eichler order Λ is not hereditary. An order is hereditary if only if all the local orders Λ is hereditary. So an Eichler Λ is hereditary if only if every k 1 (See Section 2 for the definition of k ). To prove the above results simultaneously for Eichler hereditary orders, it suffices for us to prove the result for orders Λ in A M n (D), where D is a finite dimensional algebra locally at each prime, Λ has the form ( ) ( k ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) Λ = ( ) ( ) ( ) ( k ), for some k 1, where is the unique maximal order in the division algebra D ( ) which satisfies D F F M m (D ( ) ). Such orders we shall call generalized Eichler orders (see section 2 for definitions notations). If all k are equal to 1, then Λ is hereditary (Theorem in [7]), whereas if A is a quaternion algebra, then Λ is an Eichler order as defined in [8]. In this paper, we use the same notations as in [4]. Acknowledgements: The authors are grateful to the referee for helpful comments. 2. Higher class groups of generalized Eichler orders Let F be a number field O F the ring of integers in F. Let A be a semisimple algebra over F Λ an order in A. For any maximal ideal of O F, let F, O F, A, Λ be the -completions of F, O F, A, Λ respectively. If D is a division algebra, let D ( ) be the division algebra such that D M k (D ( ) ). If R is any ring, we will denote the quotient of K m (R) by its divisible subgroup by Km(R). c Let S be the set of primes at which Λ is not maximal. As in [4], we define Cl m (Γ, S) to be the cokernel of the map K c m+1(a) S K c m+1(a ) / S K c m+1(a )/im(k c m+1(γ )), where Γ is a maximal order containing Λ. Let P S denote the set of rational primes lying under the prime ideals in S. 2

3 Lemma 2.1. The higher class group Cl 2n (Λ) is a homomorphic image of coker( S K c 2n+1(Λ ) K c 2n+1(A )). Proof. By Lemma 1.2 in [4], Cl 2n (Λ) coker( S K c 2n+1(Λ ) Cl 2n (Γ, S)). By Theorem 1 in [2], So K2n+1(A c )/im(k2n+1(γ c )) = 0. / S Cl 2n (Γ, S) = coker(k2n+1(a) c K2n+1(A c )). S Hence Cl 2n (Λ) is a homomorphic image of coker( K2n+1(Λ c ) K2n+1(A c )). S S Definition 2.2 ([7], 39.2). Let R be a ring. For each ideal I of R, let (I) m n denote the set of all m n matrices with entries in I. If {I ij : 1 i, j r} is a set of ideals in R, we write (I 11 ) (I 12 ) (I 1r ) (I 21 ) (I 22 ) (I 2r ) Λ = (I r1 ) (I r2 ) (I rr ) to indicate that Λ is the set of all matrices (T ij ) 1 i,j r, where for each pair (i, j), the matrix T ij ranges over all elements of (I ij ) n i n j. Definition 2.3. Let A M n (D), where D is a finite dimensional division algebra. We call an order Λ in A a generalized Eichler order if each Λ has the form ( ) ( k ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) Λ = ( ) ( ) ( ) ( k ), 3

4 where k 1 is the unique maximal order in D ( ). If A M ni (D i ) is a semisimple algebra, then an order Λ in A is called a generalized Eichler order if Λ Λ i, i i where Λ i is a generalized Eichler order in M ni (D i ). Proposition 2.4 ([3], Theorem A 2.2). Let R S be rings U an R S bimodule. Then the natural homomorphisms ( ) ( ) R 0 R U K n ( ) K n ( ) 0 S 0 S are isomorphisms. Let U = Hom(U, S), where U is considered as a right S-module. If R is the endomorphism ring End(U) of the right S-module U, then the natural homomorphisms ( ) ( ) R U R U K n ( ) K n ( 0 S U ) S Lemma 2.5. Let R be a ring, (R) (0) (0) (0) (R) (R) (0) (0) R 1 = (R) (R) (R) (0) R 2 =. Then the natural homomorphisms K n (R 1 ) K n (R 2 ) Proof. We will prove this lemma by induction. If r = 1, then R 1 = R 2. This is the trivial case. 4

5 Suppose that the lemma holds for r 1. Let (R) (0) (0) R (R) (R) (0) 3 = (R) (R) (R) (R) (R) (R) R (R) (R) (R) 4 = (R) (R) (R) By induction hypothesis, the homomorphisms K n (R 3) K n (R 4) (n 1,..., n r 1 ) (n 1,..., n r 1 ). Let R 3 = R 4 = ( ) R (R) (nr) R 3 (R) (nr) ( ) R (R) (nr) R 4 (R) (nr). Since K n (R 3 ) K n (R 3 ) K n ((R) (nr) ) K n (R 4 ) K n (R 4 ) K n ((R) (nr) ), the homomorphisms K n (R 3 ) K n (R 4 ) By Proposition 2.4, the homomorphisms Let K n (R 3 ) K n (R 1 ) (R) (R) (R) (0) (R) (R) (R) (0) R 5 = (R) (R) (R) (0) (R) (R) (R) (R) By Proposition 2.4, the homomorphisms K n (R 4 ) K n (R 5 ) 5.

6 Hence the compositions K n (R 5 ) K n (R 2 ) f n : K n (R 3 ) K n (R 4 ) K n (R 5 ) K n (R 2 ) Let g n be the following composition K n (R 3 ) K n (R 1 ) K n (R 2 ). Although f n g n are obtained in different ways, they are both induced by the same natural ring inclusion R 3 R 2. By the funtoriality of K-theory, f n = g n. Hence the maps g n are surjective, which implies that the maps K n (R 1 ) K n (R 2 ) Let A M n (D) be a simple algebra Λ a generalized Eichler order in A. The local order Λ is either maximal or isomorphic to some ( ) ( k ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) ( ) ( ) ( ) ( k ) where k 1 is the unique maximal order in D ( ). By the Skolem-Noether Theorem this isomorphism is given by an inner automorphism, hence there is an element a A such that ( ) ( k ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) Λ = a ( ) ( ) ( ) ( k ) 6, a 1.

7 We now define Γ = Λ, if Λ is maximal, Γ = a otherwise. By Theorem 5.3 in [7], there is a global maximal order Γ, so that Γ = Γ for all. Let I = k throughout this section, where runs through all at which Λ is not maximal. Lemma 2.6. For all n 1, the natural homomorphisms K n (Λ /IΓ ) K n (Γ /IΓ ) Proof. If Λ is maximal, then the lemma obviously holds. So we suppose that Λ is not maximal. Without loss of generality, we assume ( ) ( k ) ( k ) ( k ) ( ) ( ) ( k ) ( k ) Λ = ( ) ( ) ( ) ( k ) Γ = where is the unique maximal order in D ( ). Then (R) (0) (0) (0) (R) (R) (0) (0) Λ /IΓ = (R) (R) (R) (0) 7 a 1,

8 Γ /IΓ = where R = / k. The lemma now follows from Lemma 2.5., For any abelian group G, let G( 1 s ) be the group G Z[ 1 s ]. For any ring homomorphism f : A B, we shall write f for the induced homomorphism K n (A)( 1 s ) K n(b)( 1 s ) Lemma 2.7. For all n 1, the natural homomorphism f 1 : K n (Λ )( 1 s ) K n(γ )( 1 s ) is surjective, where s is the generator of I Z. Proof. The square Λ f 1 Γ (I). f 2 g 1 Λ /IΓ has an associated K ( 1 s ) Mayer-Vietoris sequence g 2 Γ /IΓ K n (Λ )( 1 s ) (f 1, f 2 ) K n (Γ )( 1 s ) K n (Λ /IΓ )( 1 s ) (g 1, g 2 ) K n (Γ /IΓ )( 1 s ) by [1] or [9], where for x K n (Λ )( 1 s ) (f 1, f 2 )(x) = (f 1 (x), f 2 (x)) (g1, g2 )(a, b) = g1 (a) g2 (b) for a K n (Γ )( 1 s ) b K n(λ /IΓ )( 1 s ). 8

9 For any element x K n (Γ )( 1 s ), we can find y K n(λ /IΓ )( 1 s ) such that (g 1, g 2 )(x, y) = g 1 (x) g 2 (y) = 0 by Lemma 2.6. So (x, y) ker(g 1, g 2 ) = im(f 1, f 2 ). Hence x im(f 1 ) which implies f 1 is surjective. Corollary 2.8. For all n 1, the cokernel of K n (Λ ) K n (Γ ) has no nontrivial p-torsion elements, where p is an arbitrary rational prime which does not divide s. Proof. By Lemma 2.7, the cokernel of K n (Λ ) K n (Γ ) is s-torsion. Hence the result follows. Corollary 2.9. For all n 0, the map f : K 2n+1 (Λ )( 1 s ) K 2n+1(A )( 1 s ) is surjective, where f is induced by the inclusion map f : Λ A. Proof. The map is the composition f : K 2n+1 (Λ )( 1 s ) K 2n+1(A )( 1 s ) K 2n+1 (Λ )( 1 s ) f 1 K 2n+1 (Γ )( 1 s ) h K 2n+1 (A )( 1 s ), where h is induced by the inclusion h : Γ A. By Theorem 1 of [2], h is surjective. Since f 1 h are both surjective, f is also surjective. Theorem Let Λ be a generalized Eichler order in a semi-simple algebra A over F. For all n 1, the q-primary part of Cl 2n (Λ) is trivial for q / P S. Proof. Since Λ can be expressed as the direct sum of generalized Eichler orders in the simple components of A, we may assume A is simple.. Corollary 2.9 implies in this case that coker(k c 2n+1(Λ ) K c 2n+1(A ))(q) = 0 for q / P S. So the q-primary part of Cl 2n (Λ) is trivial for q / P S by Lemma

10 References [1] R. M. Charney, A note on excision in K-theory, Algebraic K-theory, number theory, geometry analysis (Bielefeld, 1982), 47 54, Lecture Notes in Math. 1046, Springer, Berlin, [2] M. E. Keating, A transfer map in K-theory, J. London Math. Soc. (2) 16 (1977), no. 1, [3] M. E. Keating, The K-theory of triangular rings orders, Algebraic K-theory, Number theory, geometry analysis (Proceedings of the international conference held at Bielefeld, July 26 30, 1982.), , Lecture Notes in Mathematics 1046 (Springer, Berlin, 1984). [4] M. Kolster R. Laubenbacher, On higher class groups of orders, Math. Z. 228 (1998), no. 2, [5] A. O. Kuku, K n, SK n of integral group-rings orders, Contemporary Math. 55 (1986), [6] A. O. Kuku, Some finiteness results in the higher K-theory of orders group-rings, Topology Appl. 25 (1987), no. 2, [7] I. Reiner, Maximal orders, London Mathematical Society Monographs No. 5, Academic Press, London-New York, [8] M. F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics 800 (Springer, Berlin, 1980). [9] C. A. Weibel, Mayer-Vietoris sequences module structures on NK, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), pp , Lecture Notes in Math., 854, Springer, Berlin,