Vol. 74, No. 2, 1978

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Hereditary crossed product orders

Héctor Alfredo Merklen

Vol. 74 (1978), No. 2, 391–406

In this paper one deals with crossed product orders Λ of the following form: Let be a Dedekind domain with quotient field and a semisimple, commutative, algebra of finite dimension over . Let 𝒢 be a finite subgroup of the group of automorphisms of whose fixed subalgebra is , and let Λ0 be an -order in , which is 𝒢-stable. Then, if [f] is an element of the second cohomology group H2(𝒢,U0)), our order is Λ = Δ(f,Λ0,𝒢). One is interested in the set of all maximal orders in 𝒜 = Δ(f,,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 is a local complete Dedekind domain and is a Galois field extension of with group 𝒢), to the totally ramified case (i.e., when, furthermore, is totally ramified) and, finally, to the wildly ramified case.

Mathematical Subject Classification
Primary: 16A14, 16A14
Secondary: 16A18
Received: 1 November 1973
Revised: 6 May 1975
Published: 1 February 1978
Héctor Alfredo Merklen