Block-Decomposition for 3-Parameter Persistence Modules

TL;DR

This paper extends the understanding of block-decomposition in multi-parameter persistence modules by generalizing conditions from 2-parameter to 3-parameter cases, explaining the limitations of such decompositions.

Contribution

It generalizes the strong exactness condition from 2-parameter to 3-parameter persistence modules, providing insights into block-decomposability limitations.

Findings

Generalization of strong exactness to 3-parameter modules

Conditions for block-decomposition in 3-parameter modules

Explanation of why general persistence modules lack block decomposition

Abstract

In 2020, Cochoy and Oudot got the necessary and sufficient condition of the block-decomposition of 2-parameter persistence modules $\mathbb{R}^2 \to \textbf{Vec}_{\Bbbk}$. And in 2024, Lebovici, Lerch and Oudot resolve the problem of block-decomposability for multi-parameter persistence modules. Following the approach of Cochoy and Oudot's proof of block-decomposability for 2-parameter persistence modules, we rediscuss the necessary and sufficient conditions for the block decomposition of the 3-parameter persistence modules $\mathbb{R}^3 \to \textbf{Vec}_{\Bbbk}$. Our most important contribution is to generalize the strong exactness of 2-parameter persistence modules to the case of 3-parameter persistence modules. What's more, the generalized method allows us to understand why there is no block decomposition in general persistence modules to some extent.