On symmetric orders and separable algebras

Author:
T. V. Fossum

Journal:
Trans. Amer. Math. Soc. **180** (1973), 301-314

MSC:
Primary 16A16

DOI:
https://doi.org/10.1090/S0002-9947-1973-0318203-1

MathSciNet review:
0318203

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebraic number field, and let be an -order in a separable -algebra , where is a Dedekind domain with quotient field ; let denote the center of . A left -lattice is a finitely generated left -module which is torsion free as an -module. For left -modules and is a module over . In this paper we examine ideals of which are the annihilators of for certain classes of left -lattices related to the central idempotents of , and we compute these ideals explicitly if is a symmetric -algebra. For a group algebra, these ideals determine the defect of a block. We then compare these annihilator ideals with another set of ideals of which are closely related to the homological different of , and which in a sense measure deviation from separability. Finally we show that, for to be separable over , it is necessary and sufficient that is a symmetric -algebra, is separable over , and the center of each localization of at the maximal ideals of maps onto the center of its residue class algebra.

**[1]**M. Auslander and O. Goldman,*The Brauer group of a commutative ring*, Trans. Amer. Math. Soc.**97**(1960), 367-409. MR**22**#12130. MR**0121392 (22:12130)****[2]**C. W. Curtis and I. Reiner,*Representation theory of finite groups and associative algebras*, Pure and Appl. Math., vol. 11, Interscience, New York, 1962. MR**26**#2519. MR**0144979 (26:2519)****[3]**S. Endo and Y. Watanabe,*On separable algebras over a commutative ring*, Osaka J. Math.**4**(1967), 233-242. MR**37**#2796. MR**0227211 (37:2796)****[4]**T. V. Fossum,*Characters and centers of symmetric algebras*, J. Algebra**16**(1970), 4-13. MR**41**#6894. MR**0262284 (41:6894)****[5]**H. Jacobinski,*On extensions of lattices*, Michigan Math. J.**13**(1966), 471-475. MR**34**#4377. MR**0204538 (34:4377)****[6]**K. W. Roggenkamp,*Projective homomorphisms and extensions of lattices*, J. Reine Angew. Math.**246**(1971), 41-45. MR**43**#248. MR**0274485 (43:248)****[7]**K. W. Roggenkamp and V. Huber-Dyson,*Lattices over orders, I*, Lecture Notes in Math., vol. 115, Springer-Verlag, New York, 1970. MR**0283013 (44:247a)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
16A16

Retrieve articles in all journals with MSC: 16A16

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0318203-1

Article copyright:
© Copyright 1973
American Mathematical Society