Hook-decomposable modules and their resolutions

TL;DR

This paper explores various classes of biparameter persistence modules, establishing their relationships, and highlighting the structural complexity differences from monoparameter modules.

Contribution

It provides a comprehensive comparison of classes of biparameter modules, including explicit implications and counterexamples, clarifying their structural hierarchy.

Findings

$ ext{γ}$-products are a small subclass of biparameter modules

Hook-decomposable modules admit a structure theorem

Counterexamples show implications do not reverse

Abstract

We compare several classes of biparameter persistence modules: $\gamma$-products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension at most 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, $\gamma$-products (i.e., hook-decomposable modules) form a very small subclass of biparameter modules, precisely the ones for which a structure theorem still holds, thus making explicit the richer structural complexity of the biparameter setting compared to the monoparameter one.