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Persistent Leray's spectral sequence

Edivaldo L. dos Santos, Telmo I. Acosta Vellozo

Abstract

In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set and its intersections with a covering that is the pre-image under a function of a covering of a known space.

Persistent Leray's spectral sequence

Abstract

In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set and its intersections with a covering that is the pre-image under a function of a covering of a known space.

Paper Structure

This paper contains 4 sections, 3 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Persistent Cohomology
  3. Correspondence
  4. Leray's theorem

Key Result

Theorem 1.4

The correspondence $\alpha$ defines an equivalence of categories between the category of persistence modules of finite type over $R$ and the category of finitely generated graded modules over $R[t]$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (8)

  • Definition 1.1
  • Remark 1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1: Leray's Theorem
  • Theorem 2.2: Leray's Theorem for Persistence Module
  • proof