Crossed Products and Hereditary Orders

Abstract

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).