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Homological approximations in persistence theory

Benjamin Blanchette, Thomas Brüstle, Eric J. Hanson

TL;DR

It is shown that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules, and that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariants.

Abstract

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement.

Homological approximations in persistence theory

TL;DR

It is shown that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules, and that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariants.

Abstract

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement.
Paper Structure (24 sections, 31 theorems, 42 equations)

This paper contains 24 sections, 31 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Organization and main results
  3. Acknowledgements
  4. Notation and Terminology
  5. Spread modules
  6. Motivation and related invariants
  7. The dimension vector/Hilbert function
  8. The rank invariant and its generalizations
  9. Generalized Betti numbers and $g$-vectors
  10. Amplitudes and a view towards stability
  11. Relative homological algebra
  12. Approximations, exact structures, and Grothendieck groups
  13. Homological and dim-hom invariants
  14. Projectivisation
  15. Classes and morphisms of spread modules.
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\Lambda$ be a finite-dimensional algebra, and let $\mathcal{X}$ be a finite set of indecomposable $\Lambda$-modules which contains the indecomposable projectives. If the algebra $\mathop{\mathrm{\mathrm End}}\nolimits_\Lambda(\bigoplus_{R \in \mathcal{X}} R)^{op}$ has finite global dimension, t

Theorems & Definitions (89)

  • Theorem 1.1: Theorem \ref{['thm:hom_dimhom']}
  • Theorem 1.2: Theorem \ref{['thm:one_source_finite_gldim']}
  • Theorem 1.3: Theorem \ref{['thm:finer']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • ...and 79 more