Bigraded Betti numbers and generalized persistence diagrams

Abstract

Commutative diagrams of vector spaces and linear maps over $$\mathbb {Z}^2$$ Z 2 are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module M is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and Mémoli. Via the Möbius inversion of the generalized rank invariant of M, we obtain a collection of connected subsets $$I\subset \mathbb {Z}^2$$ I ⊂ Z 2 with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over $$\mathbb {R}$$ R from TDA. In this paper we show that the bigraded Betti numbers of M, a classical algebraic invariant of M, are obtained by counting the corner points of these subsets Is. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to d-parameter persistence modules for $$d \ge 3$$ d ≥ 3 .