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

Consider a weak crossed product order Cf := P σ∈G Sxσ in Σf := P σ∈G Kxσ ,w hereS is the integral closure of a discrete valuation ring R in a tamely ramified Galois extension K of the field of fractions of R. Assume that S is local. In this paper, we show that Cf is hereditary if and only if it is maximal among the weak crossed product orders in Σf . We also give an algorithm that constructs, in terms of the basis elements xσ and the cocycle f , the unique hereditary weak crossed product order in Σf that contains a given Cf , and we give a criterion for determining whether that hereditary order will have a cocycle that takes nonunit values in S.