On the representation type of local twisted group rings
type: Item
instance of: scholarly article
title: On the representation type of local twisted group rings
zbMATH Open document ID: 0546.16004
DOI: 10.1016/0021-8693(84)90198-4
author: Th. Theohari-Apostolidi
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let R be a complete discrete valuation ring with quotient field K. Let L/K be a finite Galois extension with Galois group G. Let S be the valuation ring of L with maximal ideal P. Let A be the trivial crossed- product algebra \((L/K,1)=\oplus_{\sigma\in G}L\sigma\) and let \(\Lambda\) be the twisted group ring \(S\circ G=\oplus_{\sigma\in G}S\sigma\) in which multiplication is given by \((a\sigma)(b\tau)=a\sigma (b)\sigma\tau \) (a,\(b\in L\); \(\sigma\),\(\tau \in G)\). Then A is a central simple K- algebra isomorphic to the algebra of all \(n\times n\) matrices over K with \(n=| L:K| =| G|\) and \(\Lambda\) is an R-order in A. The paper investigates the representation type of \(\Lambda\). Letting \(G_ i\) denote the ith ramification group of P in L/K it is shown that \(\Lambda\) is of finite representation type if and only if either (i) \(| G_ 1|\leq 2\) or (ii) \(| G_ 1| =3\) and \(G_ 1\neq G_ 2\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 3869536
zbMATH Keywords: crossed-product algebra
zbMATH Keywords: twisted group ring
zbMATH Keywords: central simple K-algebra
zbMATH Keywords: representation type
zbMATH Keywords: ramification group
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90198-4
OpenAlex ID: W2067690329
cites work: Ramification index and multiplicity
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cites work: Integral representations and diagrams
cites work: Artin-Schreier Equations in Characteristic Zero
cites work: Q4061092
cites work: Q5723265
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description: scientific article; zbMATH DE number 3869536