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R-orders in a Split Algebra have Finitely Many Non-Isomorphic Irreducible Lattices as soon as R has Finite Class Number

Published online by Cambridge University Press:  20 November 2018

Klaus W. Roggenkamp
Klaus W. Roggenkamp*
Affiliation:
McGill University, Montreal, Quebec
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Let R be a Dedekind domain with quotient field K and ∧ an R-order in the finite-dimensional separable K-algebra A. If K is an algebraic number field with ring of integers R, then the Jordan-Zassenhaus theorem states that for every left A-module L, the set SL(M)={M: M=∧-lattice, KM≅L} splits into a finite number of nonisomorphic ∧-lattices (cf. Zassenhaus [5]).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 3 , September 1971 , pp. 405 - 409
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Higman, D. G. and MacLaughlin, J. E., Finiteness of the class numbers of representations of algebras over function fields, Michigan Math. J. 6 (1959), 401-404. Google Scholar
2. Maranda, J. M., On the equivalence of representations of finite groups by groups of automorphisms of modules over Dedekind domains, Canad. J. Math 7 (1955), 516-526.Google Scholar
3. Reiner, I., On the class number of representations of an order, Canad. J. Math. 11 (1959), 660-672.Google Scholar
4. Roggenkamp, K. W., On irreducible lattices of orders, Canad. J. Math. 21 (1969), 970-976.Google Scholar
5. Zassenhaus, H., Neuer Beweis der Endlichkeit der Klassenzahl bei unimodularer Äquivalenz ganzzahliger Substitutionsgruppen, Hamb. Abh. 12 (1938), 276-288.Google Scholar
6. Weil, A., Basic number theory, Springer-Verlag, New York, 1967.Google Scholar