Computing Flat-Injective Presentations of Multiparameter Persistence Modules

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 $\nu F_\bullet[n]$ under the Nakayama functor are quasi-isomorphic, where $\nu F_\bullet[n]$ is a complex of injective modules. We give an explicit construction of a quasi-isomorphism $\phi_\bullet\colon F_\bullet \to \nu F_\bullet[n]$,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 $\phi_0\colon F_0 \to \nu F_n$ is a flat-injective resolution of $M$. From our construction of $\phi$, we obtain a method to compute a matrix representing $\phi_0$from the matrices representing the resolution $F_\bullet$. A Julia package implementing this method is available.