A functorial approach to monomorphism categories II: Indecomposables

Abstract

We investigate the (separated) monomorphism category $\operatorname{mono}(Q,\Lambda)$ of a quiver $Q$ over an Artin algebra $\Lambda$. We construct an epivalence from $\overline{\operatorname{mono}}(Q,\Lambda)$ to $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$, where $\operatorname{mod}\Lambda$ is the category of finitely generated modules and $\overline{\operatorname{mod}}\, \Lambda$ and $\overline{\operatorname{mono}}(Q,\Lambda)$ denote the respective injectively stable categories. Furthermore, if $Q$ has at least one arrow, then we show that this is an equivalence if and only if $\Lambda$ is hereditary. In general, it induces a bijection between indecomposable objects in $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$ and non-injective indecomposable objects in $\operatorname{mono}(Q,\Lambda)$. We show that the generalized Mimo-construction, an explicit minimal right approximation into $\operatorname{mono}{(Q,\Lambda)}$, gives an inverse to this bijection. Using this, we describe the indecomposables in the monomorphism category of a radical-square-zero Nakayama algebra, and give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length $3$ with the same residue field. These results are proved using free monads on an abelian category, in order to avoid the technical combinatorics arising from quiver representations. The setup also specializes to representations of modulations. In particular, we obtain new results on the singularity category of the algebras $H$ which were introduced by Geiss, Leclerc, and Schr\"oer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the $\iota$quivers algebras which were introduced by Lu and Wang to realize $\iota$quantum groups via semi-derived Hall algebras.