Elasticities of Orders in Central Simple Algebras

TL;DR

This paper investigates the elasticity of factorizations within orders of central simple algebras over number fields, providing criteria for finiteness and extending known results to more general non-hereditary orders.

Contribution

It characterizes when the elasticity of Hermite orders in central simple algebras is finite and extends transfer results from hereditary to non-hereditary orders.

Findings

Finiteness of elasticity characterized for quaternion orders.

Finiteness criteria established for orders in larger algebras with tiled localizations.

Extension of transfer results to non-hereditary orders.

Abstract

Let $\mathcal{O}$ be an order in a central simple algebra $A$ over a number field. The elasticitity $\rho(\mathcal{O})$ is the supremum of all fractions $k/l$ such that there exists an non-zero-divisor $a \in \mathcal{O}$ that has factorizations into atoms (irreducible elements) of length $k$ and $l$. We characterize the finiteness of the elasticity for Hermite orders $\mathcal{O}$, if either $\mathcal{O}$ is a quaternion order, or $\mathcal{O}$ is an order in an central simple algebra of larger dimension and $\mathcal{O}_{\mathfrak{p}}$ is a tiled order at every finite place $\mathfrak{p}$ at which $A_{\mathfrak{p}}$ is not a division ring. We also prove a transfer result for such orders. This extends previous results for hereditary orders to a non-hereditary setting.