On the hereditarity and maximality of weak crossed product orders over discrete valuation rings

Abstract

Consider a weak crossed product order [Formula: see text] in the crossed product algebra [Formula: see text], where [Formula: see text] is the integral closure of a discrete valuation ring [Formula: see text] in a tamely ramified Galois extension [Formula: see text] of the field of fractions of [Formula: see text]. Assume that [Formula: see text] is local. In this paper, we develop a method that can help to determine whether [Formula: see text] is a maximal order by looking for certain ideals in [Formula: see text] where [Formula: see text] is the inertia group. We demonstrate this method by classifying specific examples of hereditary weak crossed product orders as being either maximal or nonmaximal among the [Formula: see text]-orders of [Formula: see text] and we give a criterion for determining maximality of [Formula: see text] when [Formula: see text] is cyclic.