Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projective Constant Decompositions of Persistence Modules over Noetherian Rings

Nadiya Upegui Keagy

Abstract

Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the more general setting of unitary left modules over Noetherian rings. We introduce the notion of a module of projective constant type, a generalization of interval modules that facilitates decomposition results. We characterize the existence of projective constant decompositions for pointwise finitely generated persistence modules indexed by $A_n$-type quivers, totally ordered sets, and zigzag posets. Our primary results establish that such decompositions exist if and only if specific algebraic criteria are met: the projective colimit conditions (PCC) for quiver and zigzag indexings, and the projectivity of cokernels of internal morphisms for totally ordered indexings. Furthermore, we provide corollaries for the existence of classical interval decompositions over Noetherian rings where finitely generated projective modules are free, such as principal ideal domains (PIDs). Finally, we establish the uniqueness of indecomposable projective constant decompositions within the Krull-Schmidt framework and extend the uniqueness of interval decompositions to persistence modules over integral domains.

Projective Constant Decompositions of Persistence Modules over Noetherian Rings

Abstract

Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the more general setting of unitary left modules over Noetherian rings. We introduce the notion of a module of projective constant type, a generalization of interval modules that facilitates decomposition results. We characterize the existence of projective constant decompositions for pointwise finitely generated persistence modules indexed by -type quivers, totally ordered sets, and zigzag posets. Our primary results establish that such decompositions exist if and only if specific algebraic criteria are met: the projective colimit conditions (PCC) for quiver and zigzag indexings, and the projectivity of cokernels of internal morphisms for totally ordered indexings. Furthermore, we provide corollaries for the existence of classical interval decompositions over Noetherian rings where finitely generated projective modules are free, such as principal ideal domains (PIDs). Finally, we establish the uniqueness of indecomposable projective constant decompositions within the Krull-Schmidt framework and extend the uniqueness of interval decompositions to persistence modules over integral domains.
Paper Structure (9 sections, 54 theorems, 24 equations)

This paper contains 9 sections, 54 theorems, 24 equations.

Table of Contents

  1. Definitions and Notation
  2. Decomposition Criteria over Noetherian Rings
  3. Zigzag Posets
  4. TEXT-Persistence Modules
  5. Totally Ordered Persistence Modules
  6. Zigzag Persistence Modules: Finite Extrema
  7. Zigzag Persistence Modules: Infinite Extrema
  8. Necessity
  9. Uniqueness

Key Result

Theorem 1

Let $R$ be a (not necessarily commutative) Noetherian ring with identity. Let $F:A_n \to R\text{-}\textbf{Mod}$ be a nonzero pointwise finitely generated $A_n$-persistence module. Suppose that $F$ satisfies the projective colimit conditions (PCC). Then $F$ admits a projective constant decomposition.

Theorems & Definitions (134)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 124 more