Bigraded Betti numbers and generalized persistence diagrams

Commutative diagrams of vector spaces and linear maps over Z² 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⊂ Z² I ⊂ Z 2 with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over 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 I s. 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 ≥ 3 d ≥ 3 .