A Homological Approach to Factorization

TL;DR

This paper introduces a homological framework linking algebraic structures to factorization properties in domains, using functors and cochain complexes to reveal new structural insights.

Contribution

It develops a novel homological approach connecting divisibility groups and localization to factorization, providing new theorems and homology theories.

Findings

Established a functor between localizations and quotient groups

Constructed cochain complexes of po-homomorphisms

Derived fundamental structure theorems and homology insights

Abstract

Mott noted a one-to-one correspondence between saturated multiplicatively closed subsets of a domain D and directed convex subgroups of the group of divisibility D. With this, we construct a functor between inclusions into saturated localizations of D and projections onto partially ordered quotient groups of G(D). We use this functor to construct many cochain complexes of o-homomorphisms of po-groups. These complexes naturally lead to some fundamental structure theorems and some natural homology theory that provide insight into the factorization behavior of D.