🤖 AI Summary
This paper addresses the invariant characterization problem for multiparameter persistent homology modules. To tackle the structural complexity of persistence modules over finite and infinite posets, we introduce an exact category framework coupled with relative homological algebra, systematically extending classical representation-theoretic methods. Innovatively, we construct a novel adjoint pair, enabling—for the first time—the adaptation of Auslander–Reiten-type arguments to infinite posets lacking AR sequences; we explicitly classify irreducible morphisms between relative projective modules across multiple exact structures. Theoretically, we establish an equivalence between the global dimension of an exact structure and the representation dimension of the associated poset algebra. Applicationally, we prove that finitely presented modules over the plane admit no upset decomposition, whereas such decompositions exist for modules over the closed positive quadrant. These results provide a unified, computable invariant theory underpinning the algebraic foundations of multiparameter persistent homology.
📝 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.