Combinatorial resolutions of multigraded modules and multipersistent homology

TL;DR

This paper introduces a combinatorial approach to resolving multigraded modules by interpreting them as multipersistent homology modules, utilizing cellular resolutions to reflect their combinatorial structure.

Contribution

It provides a novel multigraded resolution framework for multipersistent homology modules using cellular resolutions of monomial ideals.

Findings

Provides a cellular resolution construction for multipersistent modules

Connects multipersistence to combinatorial cellular complexes

Offers insights into the acyclicity defect of multigraded chain complexes

Abstract

Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and reflects the combinatorial structure of multipersistence homology modules. In the one critical case, a multifiltration is represented by a labelled cellular complex. A multipersistence homology module measures the defect of acyclicity of the associated multigraded cellular chain complex.