A note on cancellation in totally definite quaternion algebras
type: Item
instance of: scholarly article
title: A note on cancellation in totally definite quaternion algebras
zbMATH Open document ID: 1329.11122
author: Daniel Smertnig
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2015-11-11T00:00:00Z
review text: In [J. Reine Angew. Math. 286/287, 257--277 (1976; Zbl 0368.16002)], \textit{M.-F. Vignéras} showed that there exist only finitely many isomorphism classes of Eichler orders of square-free level in totally definite quaternion algebras over number fields having locally free cancellation. She provided a criterion (called Vignéras' criterion in what follows), determined that locally free cancellation is possible only over number fields of degree at most \(33\), and classified the Eichler orders of square-free level having locally free cancellation in the case of quadratic and cubic abelian fields. Using Vignéras' criterion and some more recent results, \textit{E. Hallouin} and \textit{C. Maire} [J. Reine Angew. Math. 595, 189--213 (2006; Zbl 1100.15007)] completed the classification in the general cases by first improving the degree bound to \(8\) and analyzing the remaining cases. In the paper under review, the author mentions that there is an error in this classification proposed in the previous articles, and he corrects it. More precisely, as the author remarks, a first reduction step used to prove Vignéras' criterion says that an order \(\mathcal{O}\) in a quaternion algebra over a number field has locally free cancellation if and only if every stably free left \(\mathcal{O}\)-module is free (cf. the Introduction of [Zbl 0368.16002]). The author mentions that while locally free cancellation implies that every stably free left \(\mathcal{O}\)-module is free, the converse is false. The author explains this in Section 2. In Section 3, the author gives an explicit example of a maximal order \(\mathcal{O}\) that has the property that every stably free left \(\mathcal{O}\)-module is free, but that does not have locally free cancellation. The reviewer notes that at the end of Section 3, the equation \([\mathcal{O}_1^*\mathcal{W}_1:A^*]=2\) apparently has a misprint, and should be replaced by \([\mathcal{O}_1^*:\mathcal{W}_1A^*]=2\). Section 4 provides the revised classifi…
reviewed by: Cherng-tiao Perng
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 16H10
zbMATH DE Number: 6507152
zbMATH Keywords: quaternion algebra
zbMATH Keywords: Eichler order
MaRDI profile type: Publication
OpenAlex ID: W2327014554
DOI: 10.1515/CRELLE-2013-0069
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description: scientific article; zbMATH DE number 6507152