Computing Flat-Injective Presentations of Multiparameter Persistence Modules
Fabian Lenzen
TL;DR
The paper develops a constructive framework to obtain flat-injective presentations of multiparameter persistence modules by exploiting a Nakayama-duality-based quasi-isomorphism $\tilde{\varphi}_\bullet: F_\bullet \to \nu F_\bullet[n]$ for a finite-rank free resolution $F_\bullet$. It provides an explicit, cubic-time procedure to compute the degree-zero map $\varphi_0$, which yields a flat-injective presentation $\varphi: F_0 \to I^0$ of the module $M$, and packages this into a practical algorithm with a Julia implementation FlangePresentations.jl. The core construction uses a Čech-type flat resolution $\tilde{\Omega}_\bullet$ and a totalization of a double complex to realize the quasi-inverse, with concrete formulas for contraction and boundary data. The approach enables efficient computation of flat-injective presentations and enables downstream invariants computation (e.g., rank invariant, persistence images) for multiparameter persistence, and extends to computing presentations of persistent-homology modules, demonstrated via examples and software.
Abstract
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 $φ_\bullet\colon F_\bullet \to νF_\bullet[n]$,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 $φ_0\colon F_0 \to νF_n$ is a flat-injective resolution of $M$. From our construction of $φ$, we obtain a method to compute a matrix representing $φ_0$from the matrices representing the resolution $F_\bullet$. A Julia package implementing this method is available.