Hyperplane Restrictions of Indecomposable $n$-Dimensional Persistence Modules

TL;DR

This paper extends the understanding of the structure of indecomposable n-dimensional persistence modules by showing they can be constructed as hyperplane restrictions of higher-dimensional modules, with applications to zigzag modules.

Contribution

It generalizes previous results by proving that any finitely presented (n-1)-dimensional persistence module with finite support is a hyperplane restriction of an indecomposable n-dimensional module.

Findings

Any finitely presented (n-1)-dimensional module is a hyperplane restriction of an indecomposable n-dimensional module.

Finite zigzag persistence modules can be realized as restrictions of indecomposable 3-dimensional modules.

The results deepen the structural understanding of multipersistence modules.

Abstract

Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular $(n-1)$-dimensional persistence module with finite support is a hyperplane restriction of an $n$-dimensional persistence module. We extend this result to the following: If $M$ is any finitely presented $(n-1)$-dimensional persistence module with finite support, then there exists an indecomposable $n$-dimensional persistence module $M'$ such that $M$ is the restriction of $M'$ to a hyperplane. We also show that any finite zigzag persistence module is the restriction of some indecomposable $3$-dimensional persistence module to a path.