Retrieving biparameter persistence modules from monoparameter ones: a characterization of hook-decomposable persistence modules

TL;DR

This paper introduces a product construction linking biparameter and monoparameter persistence modules, revealing a structural characterization of hook-decomposable modules, which enhances understanding of their relationships.

Contribution

It presents a novel product construction that relates biparameter modules to monoparameter modules and characterizes hook-decomposable modules structurally.

Findings

The construction effectively relates biparameter and monoparameter modules.

It characterizes hook-decomposable modules structurally.

The approach provides new insights into the structure of persistence modules.

Abstract

Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of product between two monoparameter persistence modules. While originally conceived to serve this comparative purpose, our construction unexpectedly reveals a deeper structural property: it also characterizes a class of biparameter modules known as hook-decomposable modules.