Dualities in Multiparameter Persistence

TL;DR

This paper generalizes a duality in persistent homology to multiparameter settings, providing theoretical foundations and an algorithm for computing free resolutions of multiparameter persistence modules.

Contribution

It introduces a multiparameter duality for persistence modules, extending classical one-parameter duality, with proofs and implications for free resolution computations.

Findings

Establishes a multiparameter duality functor relationship.

Provides an elementary proof and a Grothendieck duality interpretation.

Lays groundwork for an algorithm to compute free resolutions in multiparameter persistence.

Abstract

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual $(-)^*$ and the global dual $(-)^\dagger$, we show that $H_q(C)^* \cong H^{N+q}(C^\dagger)$ for chain complexes $C$ of free $N$-parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module $M$ and those of its pointwise dual $M^*$, a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.