On Projective and Flat Persistence Modules

TL;DR

This paper investigates conditions under which persistence modules indexed by lattices or preordered sets are projective, free, or flat, and provides algorithms for computing bases of free modules in specific cases.

Contribution

It offers new criteria for projectivity and flatness of persistence modules over various indexing sets, and introduces algorithms for basis computation in specific lattice cases.

Findings

Criteria for projective modules to be free over a PID with lattice indexing

Conditions under which persistence modules are not projective when indexed by a lattice

Algorithms for computing bases of free persistence modules over  and ^2

Abstract

In recent years, persistence modules have been viewed as graded modules with gradation over a preordered set serving as the indexing set. We provide sufficient criteria for a projective module over a PID to be free when the indexing set is a lattice. With a lattice as the indexing set, we obtain criteria ensuring that a given persistence module is not projective. When the indexing set is a preordered set, we establish the flatness of a well-known family of persistence modules. We end the article with two algorithms to compute a basis of free persistence modules with indexing sets $\mathbb{Z}$ and $\mathbb{Z}^2$.