Definite orders with locally free cancellation

Abstract

We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.

Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

“…Let ζ K psq be the Dedekind zeta function, let h K " |CpO K q| be the class number of K and let ∆ K be the discriminant of K. The following was proven in [14]. The following was first shown by Vignéras in [43], though a simplified proof can be found in [36,Theorem 5.11]. Theorem 4.2.…”

“…Define the class set Cls Λ as the set of isomorphism classes of rank one locally free Λ-modules, which is finite by the Jordan-Zassenhaus theorem [6, Section 24]. Equivalently, this is the set of locally principal fractional Λ-ideals, under the relation I " J if there exists α P A ˆsuch that I " αJ (see [36]). This comes with the stable class map r ¨sΛ : Cls Λ Ñ CpΛq which sends P Þ Ñ rP s and is surjective since every locally free Λ-module P is of the form P 0 ' Λ i where P 0 P Cls Λ and i ě 0 [15].…”

“…It is possible to show that Λ 2n has stably free cancellation in all other cases. It can be shown using[36, Table…”

Finiteness of elasticities of orders in central simple algebras

“…Thus the only non-Hermite orders appear in definite quaternion algebras. Restrict now to the case where R = R S is the ring of algebraic integers of the center of K. Then there exists a complete classification of the definite Hermite quaternion orders [SV19]. If O is a maximal R-order that is not Hermite, it is known that there cannot be a transfer homomorphism to a monoid of zero-sum sequences [Sme13, Theorem 1.2].…”