Projective Constant Decompositions of Persistence Modules over Noetherian Rings

TL;DR

This paper generalizes the decomposition theory of persistence modules from fields to Noetherian rings, introducing projective constant modules and establishing conditions for their existence and uniqueness across various indexing posets.

Contribution

It introduces the concept of projective constant modules for persistence modules over Noetherian rings and characterizes their existence and uniqueness for different indexings.

Findings

Existence of decompositions characterized by algebraic criteria

Corollaries for classical interval decompositions over PIDs

Uniqueness of indecomposable decompositions in Krull-Schmidt framework

Abstract

Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the more general setting of unitary left modules over Noetherian rings. We introduce the notion of a module of projective constant type, a generalization of interval modules that facilitates decomposition results. We characterize the existence of projective constant decompositions for pointwise finitely generated persistence modules indexed by $A_n$-type quivers, totally ordered sets, and zigzag posets. Our primary results establish that such decompositions exist if and only if specific algebraic criteria are met: the projective colimit conditions (PCC) for quiver and zigzag indexings, and the projectivity of cokernels of internal morphisms for totally ordered indexings. Furthermore, we provide corollaries for the existence of classical interval decompositions over Noetherian rings where finitely generated projective modules are free, such as principal ideal domains (PIDs). Finally, we establish the uniqueness of indecomposable projective constant decompositions within the Krull-Schmidt framework and extend the uniqueness of interval decompositions to persistence modules over integral domains.