2024/01/11 by Fabian Lenzen, Lenzen, Fabian
Computer Science · Biochemistry, Genetics and Molecular Biology · #Topological and Geometric Data Analysis #Cell Image Analysis Techniques #Data Visualization and Analytics
paper · pdf · doi:10.48550/arxiv.2401.06008
A flat-injective presentation of a multiparameter persistence module M characterizes M as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be easily represented by a single graded matrix, completely describe the persistence module up to isomorphism, and can be used as starting point to compute other invariants of it,such as the rank invariant, persistence images, and others. If all homology modules of a bounded chain complex F_\bullet of flat n-parameter modules are finite dimensional,it is known that F_\bullet and its shifted image νF_\bullet[n] under the Nakayama functor are quasi-isomorphic, where νF_\bullet[n] is a complex of injective modules. We give an explicit construction of a quasi-isomorphism ϕ_\bullet\colon F_\bullet → νF_\bullet[n],based on the boundary morphisms of F_\bullet. If F_\bullet is a flat resolution of a finite dimensional persistence module M,then the degree-zero part ϕ0\colon F0 → νFn is a flat-injective resolution of M. From our construction of ϕ, we obtain a method to compute a matrix representing ϕ0from the matrices representing the resolution F_\bullet. A Julia package implementing this method is available.