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In this paper one deals with
crossed product orders Λ of the following form: Let ℛ be a Dedekind domain with
quotient field ℱ and ℰ a semisimple, commutative, algebra of finite dimension over
ℱ. Let 𝒢 be a finite subgroup of the group of automorphisms of ℰ whose fixed
subalgebra is ℱ, and let Λ0 be an ℛ-order in ℰ, which is 𝒢-stable. Then, if [f]
is an element of the second cohomology group H2(𝒢,U(Λ0)), our order is
Λ = Δ(f,Λ0,𝒢). One is interested in the set of all maximal orders in 𝒜 = Δ(f,ℰ,𝒢)
which contain Λ and also in all hereditary orders in 𝒜 which contain Λ.
In particular, one is interested in knowing sufficient conditions for Λ itself
to be hereditary. This last question is answered by Theorem 1, and the
other, more general question, is succesively reduced to the classical complete
case (i.e., when ℛ is a local complete Dedekind domain and ℰ is a Galois
field extension of ℱ with group 𝒢), to the totally ramified case (i.e., when,
furthermore, ℰ∕ℱ is totally ramified) and, finally, to the wildly ramified
case.
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