Crossed product and hereditary orders

Gerald Howard Cliff and Alfred Rheinhold Weiss

Vol. 122 (1986), No. 2, 333–345

DOI: 10.2140/pjm.1986.122.333

Abstract

Let Λ be the crossed product order (O_L∕O_K,G,ρ) where L∕K is a finite Galois extension of local fields with Galois group G, and ρ is a factor set with values in O_L^∗. Let Λ₀ = Λ, and let Λ_i+1 be the left order O_l(rad Λ_i) of rad Λ_i. The chain of orders Λ₀,Λ₁,…,Λ_s ends with a hereditary order Λ_s. We prove that Λ_s is the unique minimal hereditary order in A = KΛ containing Λ, that Λ_s has e∕m simple modules, each of dimension f over the residue class field K of O_K, and that s = d − (e − 1). Here d, e, f are the different exponent, ramification index, and inertial degree of L∕K, and m is the Schur index of A.

Mathematical Subject Classification 2000

Primary: 16A18, 16A18

Secondary: 11S45

Milestones

Received: 21 September 1984

Revised: 26 March 1985

Published: 1 April 1986