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

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

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.

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

Paper Structure

This paper contains 1 section, 4 theorems, 6 equations, 2 figures.

Table of Contents

  1. Acknowledgements

Key Result

Proposition 1

$M_f \MVAt_{\gamma} M_g$ is a finitely generated $\mathbb{N}^2$-graded $\mathbb{F}[x,y]$-module.

Figures (2)

  • Figure 1: A simplicial complex $X$ and two functions $f, g$ defined simplex-wise on it.
  • Figure 2: Comparison of the supports of (a) $M_f \MVAt_{\gamma_{\mathrm{bott}}} M_g$ and (b) $M_{(f,g)}$.

Theorems & Definitions (9)

  • Definition 1: $\gamma$-product of $M_f$ and $M_g$
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Definition 2
  • Example 1
  • Definition 3: botnan2024signedbotnan2024bottleneck
  • Proposition 3
  • Proposition 4