The Characterization of Finite Elasticities

Abstract

Our main motivating goal is the study of factorization in Krull Domains H with finitely generated class group G. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. One of the most important is the elasticity ρ(H) = lim k→∞ ρ k (H)/k, where ρ k (H) is the maximal number of atoms in any re-factorization of a product of k atoms. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. The elasticities, as well as many other arithmetic invariants, are the same as those of an associated combinatorial monoid B(G0) of zero-sum sequences, where G0 ⊆ G are the classes containing height one primes.We characterize when finite elasticity holds for any Krull Domain with finitely generated class group G. Indeed, our results are valid for the more general class of Transfer Krull Monoids (over a subset G0 of a finitely generated abelian group G). Moreover, we show there is a minimal s ≤ (d + 1)m, where d is the torsion free rank and m is the exponent of the torsion subgroup, such that ρsOur characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset G ⋄ 0 ⊆ G0 and global bound N such that there are no nontrivial zero-sum sequences with terms from G ⋄ 0 , and every minimal zero-sum sequence has at most N terms from G0 \ G ⋄ 0 . We give an explicit description of G ⋄ 0 in terms of the Convex Geometry of G0 modulo the torsion subgroup GT ≤ G, and show that finite elasticity is equivalent to there being no positive R-linear combination of the elements of this explicitly defined subset G ⋄ 0 equal to 0 modulo GT . Additionally, we use our results to show finite elasticity implies the set of distances ∆(H), the catenary degree c(H) (for Krull Monoids) and a weakened form of the tame degree (for Krull Monoids) are all also finite, and that the Structure Theorem for Unions holds-four of the most commonly used measurements of structured factorization, after the elasticity.Our results for factorization in Transfer Krull Monoids are accomplished by developing an extensive theory in Convex Geometry generalizing positive bases. The convex cone generated byPositive bases were first introduced and studied in the mid 20th century, and the structural work initiated by Reay led to a simplified proof and strengthening of Bonnice and Klee's celebrated generalization of Carathéordory's Theorem. They have since found increasing importance in areas of applied mathematics. We show that the structural result of Reay can be extended to special types of complete simplicial fans, which we term Reay systems. We extend these results to a general theory dealing with infinite sequences of Reay systems, as well as their limit structures. The latter, while more complex, avoid the introduction of linear dependencies into the limit structure that were not originally present, circumventing the general obstacle that a limit of linearly independent sets can degenerate into linear...