Relative preprojective partitions (Q1911777)

Let \(\Lambda\) be an Artin algebra, the notion of postprojective, or preprojective, partition in the category \(\text{mod }\Lambda\) of finitely generated right \(\Lambda\)-modules was introduced by \textit{M. Auslander} and \textit{S. O. Smalø} [in J. Algebra 66, 61-122 (1980; Zbl 0477.16013)]. In the present paper, the author introduces and studies the properties of postprojective partitions relative to some epiclass, by which is meant a class of morphisms enjoying some of the properties (and therefore replacing) the class of epimorphisms of \(\text{mod }\Lambda\). Given such an epiclass \(\mathcal E\), the author defines splitting \(\mathcal E\)-projectives, \(\mathcal E\)-covers, \(\mathcal E\)-postprojective partitions and modules. In this context, he derives existence results and basic properties of \(\mathcal E\)-postprojective modules. He then considers the case where \(\mathcal E\) is induced by a covariant functor \(F:\text{mod }\Lambda\to Ab\) in the sense that \(\mathcal E\) consists of all \(f\) such that \(Ff\) is an epimorphism. In this case, the \(\mathcal E\)-postprojective modules can be described by means of the radical series of \(F\) while, conversely, the relative postprojective partition yields a filtration of the functor \(F\). The author also studies relations between relative partitions induced by distinct epiclasses, and relative partitions determined by suitable quasi-projective modules.