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Hook-decomposable modules and their resolutions

Isabella Mastroianni, Marco Guerra, Ulderico Fugacci, Emanuela De Negri

Abstract

We compare several classes of biparameter persistence modules: $γ$-products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, $γ$-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.

Hook-decomposable modules and their resolutions

Abstract

We compare several classes of biparameter persistence modules: -products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, -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.
Paper Structure (3 sections, 2 theorems, 2 figures)

This paper contains 3 sections, 2 theorems, 2 figures.

Table of Contents

  1. Motivation and contribution
  2. Background notions
  3. Hook-decomposability: sufficient and necessary conditions

Key Result

Theorem 3.1

(Characterization of hook-decomposability) The following are equivalent: $(i)\,M$ is a $\gamma$-product; $(ii)\,M$ is hook-decomposable; $(iii)$ the structure theorem holds for $M$.

Figures (2)

  • Figure 1: Implication diagram that we establish in this work.
  • Figure 2: Visual representation of the studied classes. Examples of possible supports are shown for each one. Lines: green = births; red-dotted = deaths; orange-dashed = merges. Blue areas = supports interior.

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 3.1
  • proof : Sketch of the proof
  • Theorem 3.2
  • proof : Sketch of the proof
  • Remark 3.3