Exact structures for persistence modules

TL;DR

This paper explores the application of exact structures and relative homological algebra to invariants of multiparameter persistence modules, introducing new results on lifting arguments to infinite posets and describing irreducible morphisms.

Contribution

It presents a novel adjunction for lifting Auslander-Reiten theory to infinite posets and explicitly describes irreducible morphisms between relative projective modules.

Findings

Established a new adjunction for infinite posets

Provided examples of resolutions by upsets in specific settings

Described irreducible morphisms for various exact structures

Abstract

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.