Hereditary crossed product orders over discrete valuation rings

In this paper, we consider weak crossed product orders Af=∑Sxσ with coefficients in the integral closure S of a discrete valuation ring R in a tamely ramified Galois extension of the field of fractions of R. In the first section, we compute the Jacobson radical of Af when S is local, and we give a characterization of the hereditarity of the order in terms of the cocycle values. In the second section, we prove (again in the local case) that every σ in the inertia group for S/R must belong to {σ∈G|f(σ,σ−1) is a unit of S}. In the final section, we compute the Jacobson radical in the general case (S is semilocal) and show how the hereditarity of Af can be determined locally under an additional hypothesis.