  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 theory is not available. 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.

Algebraic Topology;Computational Geometry;Representation Theory