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.

翻译：本文讨论了恰当结构和相对同调代数在多参数持久模不变量研究中的应用。本文主要为综述性质，但包含两项新结果。对于有限偏序集，关于恰当结构中相对投射模的经典论证利用了Auslander-Reiten理论。我们的一项新结果建立了一个新的伴随关系，使得这些论证能够"提升"到某些无法使用Auslander-Reiten理论的无限偏序集上。我们给出了这一提升过程的若干例子，特别强调了在分别处理平面与正象限闭包的有有限表示时，上集分解的不存在性与存在性。随后我们将注意力限制在有限偏序集上。在此框架下，我们讨论了恰当结构的全局维数与偏序集关联代数的表示维数之间的关系。最后我们呈现第二项新贡献：对文献中已出现的若干恰当结构中相对投射模之间的不可约态射给出了显式描述。