The Characterization of Finite Elasticities

Our motivating goal is factorization in Krull Domains $H$ with finitely generated class group $G$. The elasticity $\rho(H)$ is the maximal number of atoms in any re-factorization of a product of $k$ atoms. The elasticities are the same as those of a combinatorial monoid of zero-sum sequences $B(G_0)$, where $G_0\subseteq G$ are the classes with height one primes. We characterize when finite elasticity holds for any Krull Domain with finitely generated class group. Our results are valid for the more general class of Transfer Krull Monoids (over a subset $G_0$ of a finitely generated abelian group $G$). We show there is a minimal $s\leq (d+1)m$, where $d$ is the torsion free rank and $m$ is the torsion exponent, such that $\rho_s(H)<\infty$ implies $\rho_k(H)<\infty$ for all $k\geq 1$. This ensures $\rho(H)<\infty$ if and only if $\rho_{(d+1)m}(H)<\infty$. Our characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset $G_0^\diamond\subseteq G_0$ and bound $N$ such that there are no nontrivial zero-sum sequences with terms from $G_0^\diamond$, and every minimal zero-sum sequence has at most $N$ terms from $G_0\setminus G_0^\diamond$. We give an explicit description of $G_0^\diamond$ in terms of the Convex Geometry of $G_0$ modulo the torsion subgroup $G_T\leq G$, and show finite elasticity is equivalent to there being no positive linear combination of the elements of this explicitly defined subset equal to $0$ modulo $G_T$. We use our results to show finite elasticity implies the set of distances $\Delta(H)$, the catenary degree $\mathsf c(H)$ (for Krull Monoids) and a weakened tame degree (for Krull Monoids) are all also finite, and that the Structure Theorem for Unions holds. Our results for factorization in Transfer Krull Monoids are accomplished by developing an extensive theory in Convex Geometry generalizing positive bases.