Crossed-products orders over discrete valuation rings
type: Item
instance of: scholarly article
title: Crossed-products orders over discrete valuation rings
zbMATH Open document ID: 0609.16004
DOI: 10.1016/0021-8693(87)90182-7
author: Darrell Haile
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The generalized cohomology theory of \textit{D. E. Haile}, \textit{R. G. Larson} and \textit{M. E. Sweedler} [Am. J. Math. 105, 689-814 (1983; Zbl 0523.13005)] is applied to the construction of crossed-product orders analogous to the classical crossed-product algebras. Let R be a discrete valuation ring with field of fractions F, K/F a finite Galois extension with group G and S the integral closure of R in K. S/R is assumed to be unramified. For any 2-cocycle \(f: G\times G\to S^*\) the author forms a crossed-product order \(A_ f=\coprod_{\sigma \in G}Sx_{\sigma}\) with the usual rules of multiplication which is an order in the crossed- product algebra \(\Sigma_ f=\coprod_{\sigma \in G}Kx_{\sigma}.\) From the author's introduction: ''In the first section we derive some basic properties of the cocycles and the orders. We show how to associate a finite graph to each cocycle. One of the themes of the paper is the relationship between properties of this graph and the structure of the order. Also in this section we show that if the residue field is perfect, then every maximal order is equivalent to a crossed-product order. In Section 2 we assume that S is a discrete valuation ring. In this case the crossed-product orders are primary and there is a simple characterization of those cocycles which give rise to maximal orders. Also in this section we show how to determine the ideals in the order and give necessary and sufficient conditions for two orders \(A_{f_ 1}\) and \(A_{f_ 2}\) to be isomorphic as R-algebras. In Section 3 we take up the general case (S/R unramified but not necessarily local). Let M be a maximal ideal of S with decomposition group \(D_ M\). If \(f: G\times G\to S\) is a cocycle, then \(A_ f\) is primary if and only if there are coset representatives \(g_ 1,...,g_ r\) of \(D_ M\) in G with \(f(g_ i,g_ i^{-1})\not\in M\) for all i. If \(f_ M: D_ M\times D_ M\to S\) denotes the restriction of f, then we can form the new crossed-product order \(A_ f\). If \…
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 3985440
zbMATH Keywords: generalized cohomology theory
zbMATH Keywords: crossed-product orders
zbMATH Keywords: crossed-product algebras
zbMATH Keywords: discrete valuation ring
zbMATH Keywords: finite Galois extension
zbMATH Keywords: 2-cocycle
zbMATH Keywords: maximal order
zbMATH Keywords: ideals
zbMATH Keywords: primary orders
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90182-7
OpenAlex ID: W1975582047
cites work: Q5573504
cites work: Maximal Orders
cites work: The Brauer Group of a Commutative Ring
cites work: Separable algebras over commutative rings
cites work: A New Invariant for C Over R: Almost Invertible Cohomology Theory and the Classification of Idempotent Cohomology Classes and Algebras by Partially Ordered Sets with a Galois Group Action
cites work: Some criteria for hereditarity of crossed products
cites work: Q4061092
description: scientific article; zbMATH DE number 3985440