Local characterization of block-decomposability for multiparameter persistence modules

TL;DR

This paper extends local decomposability conditions for persistence modules from 2-parameter to n-parameter settings, providing a general structure theorem applicable to modules over products of ordered sets.

Contribution

It generalizes the local condition for block module decomposability to n-parameter persistence modules and proves a comprehensive structure theorem.

Findings

Extended the local decomposability condition to n-parameter modules.

Proved a structure theorem for modules over finite products of ordered sets.

Generalized proof techniques from 2 to n parameters.

Abstract

Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a prominent role in levelset persistence. Here we generalize the local condition for decomposability into block modules to the n-parameter setting, and prove a corresponding structure theorem. Our result holds in the generality of pointwise finite-dimensional modules over finite products of arbitrary totally ordered sets. Our proof extends the one by Botnan and Crawley-Boevey from 2 to n parameters, which requires some crucial adaptations at places where their proof is fundamentally tied to the 2-parameter setting.