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On Projective and Flat Persistence Modules

Prateep Chakraborty, Giriraj Ghosh

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 $\mathbb{Z}^2$.

On Projective and Flat Persistence Modules

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 and .
Paper Structure (14 sections, 27 theorems, 54 equations, 1 figure, 3 algorithms)

This paper contains 14 sections, 27 theorems, 54 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preordered sets and posets
  4. Graded projective and graded flat modules
  5. Persistence modules as graded modules
  6. Monoid ring:
  7. Graded module structure of a persistence module:
  8. Some necessary results
  9. Basis for the projective module
  10. Proof of the Theorem \ref{['0']}:
  11. A Family of flat persistence module, which are not projective
  12. Algorithms
  13. When the indexing set is Z
  14. When the indexing set is Z2

Key Result

Theorem 1.2

Let $R$ be a principal ideal domain, and let $P$ be a lattice equipped with a compatible abelian group structure. Suppose with the indexing set $P$, $M$ is a persistence module over $R$ such that and each $M_a$ is finitely generated $R$-module. Then $M$ is graded projective if and only if it is graded free. (See Section sec2 for a detailed discussion of the necessary concepts.)

Figures (1)

  • Figure 1: Flat but not projective modules

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1
  • proof
  • ...and 59 more