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：我们讨论了恰当结构与相对同调代数在多参数持久模不变量研究中的应用。本文主要为综述性文章，但包含两个新结果。在有限偏序集上，关于恰当结构的相对投射模的经典论证利用了Auslander-Reiten理论。我们的一个结果建立了一个新的伴随关系，使得我们可以将这些论证“提升”到某些不适用Auslander-Reiten理论的无限偏序集上。我们给出了这一提升的若干实例，特别强调了在处理平面与正象限闭包上的有限可表示表示时，分别存在与不存在由“上集”构成的消解。随后，我们将注意力限制在有限偏序集上。在此设定下，我们讨论了恰当结构的整体维数与偏序集的关联代数的表示维数之间的关系。最后，我们介绍了第二个新贡献：针对文献中已出现的若干恰当结构，给出了相对投射模之间不可约态射的显式描述。