Approximation by interval-decomposables and interval resolutions of persistence modules

TL;DR

This paper explores new homological methods for approximating two-parameter persistence modules in topological data analysis, establishing finite global dimension for interval resolutions and linking different approximation approaches.

Contribution

It introduces homological interval resolutions, proves their finite global dimension for finite posets, and connects interval approximations with interval resolutions in specific cases.

Findings

Interval resolution global dimension is finite for finite posets.

A formula links interval approximations and interval resolutions in the commutative ladder case.

The maximum of interval dimensions of Auslander-Reiten translates determines the global dimension.

Abstract

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.