A note on cancellation in totally definite quaternion algebras (Q890757)

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 classification: in 4.1 the author corrected the errors and supplemented the missing invariants in the classifications of [Zbl 0368.16002] and [Zbl 1100.15007]; in 4.2 the author provides a list of invariants of Eichler orders \(\mathcal{O}\) of square-free level that do not have locally free cancellation, but that do have the property that every stably free left \(\mathcal{O}\)-module is free.