A functorial approach to monomorphism categories II: Indecomposables

Abstract We investigate the (separated) monomorphism category of a quiver over an Artin algebra . We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from to , where is the category of finitely generated ‐modules and and denote the respective injectively stable categories. Furthermore, if has at least one arrow, then we show that this is an equivalence if and only if is hereditary. In general, the epivalence induces a bijection between indecomposable objects in and noninjective indecomposable objects in , and we show that the generalized Mimo‐construction, an explicit minimal right approximation into , gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radical square zero Nakayama algebras, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. The main tool to prove these results is the language of a free monad of an exact endofunctor on an arbitrary abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras that were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the algebras that were introduced by Lu and Wang to realize groups via semiderived Hall algebras.