Crossed Products and Maximal Orders

Abstract

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

Equivalence Classes of Maximal Orders

“…Finally we mention that the equivalence relation on the set of maximal orders over R is induced by the Brauer group of the quotient field k of R. That is to say, if Σi an d Σ2 are equivalent central simple algebras over the quotient field of a discrete rank one valuation ring, then the maximal orders of Σi are equivalent to the maximal orders of Σ 2 (see Lemma 2. 1 of [11]). …”

“…2. 4 of [11]). We may therefore assume that a is in U[U) where U denotes the inertia ring of L over k.…”

“…Since Σ is in T(fe) we may consider a field L t satisfying the conclusion of Lemma 4.1. Theorem 2.3 of [11] now implies that a maximal order in J{f, L, G) is equivalent to a crossed product over a tamely ramified extension of R .…”

“…2. 3 of [11] that a maximal order in Σ is equivalent to a crossed product over a tamely ramified extension of R.…”

“…In this section we prove the main theorem of the paper, namely that a maximal order in a central simple fc-algebra Σ is equivalent to a crossed product over a tamely ramified extension of R if and only if Σ is in T{k). Both the necessity and sufficiency parts of the proof depend upon the main theorem on crossed products and maximal orders presented by the author in [11].…”