## Arithmetical invariants of local quaternion orders

### Volume 186 / 2018

#### Abstract

Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$.
The *elasticity* of $R^\bullet$ is $\rho(R^\bullet) = \sup\{k/l : u_1\cdots u_k = v_1 \cdots v_l$ with $u_i, v_j$ atoms of $R^\bullet$ and $k, l \ge 1\}$ and is one of the basic arithmetical invariants that is studied in factorization theory.
We characterize finiteness of $\rho(R^\bullet)$ and show that the set of distances $\Delta(R^\bullet)$ and all catenary degrees $\mathsf c_{\mathsf d}(R^\bullet)$ are finite.
In the setting of non-commutative orders in central simple algebras, such results have only been known for hereditary orders and for a few individual examples.