Constructing the hereditary crossed product order containing a given weak crossed product order and a criterion for weakness

Abstract

Consider a weak crossed product order [Formula: see text] in [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 show that [Formula: see text] is hereditary if and only if it is maximal among the weak crossed product orders in [Formula: see text]. We also give an algorithm that constructs, in terms of the basis elements [Formula: see text] and the cocycle [Formula: see text], the unique hereditary weak crossed product order in [Formula: see text] that contains a given [Formula: see text], and we give a criterion for determining whether that hereditary order will have a cocycle that takes nonunit values in [Formula: see text].