A functorial approach to monomorphism categories. II: Indecomposables. (English) Zbl 07948142

Gao, Nan; Külshammer, Julian; Kvamme, Sondre; Psaroudakis, Chrysostomos

Proc. Lond. Math. Soc. (3) 129, No. 4, Article ID e12640, 61 p. (2024).

Summary: We investigate the (separated) monomorphism category \(\mathrm{mono}(Q, \Lambda)\) of a quiver over an Artin algebra \(\Lambda\). We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from \(\overline{\mathrm{mono}}(Q, \Lambda)\) to \(\mathrm{rep}(Q, \overline{\bmod}\Lambda)\), where \(\bmod \Lambda\) is the category of finitely generated \(\Lambda\)-modules and \(\overline{\bmod}\Lambda\) and \(\overline{\mathrm{mono}}(Q, \Lambda)\) 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, the epivalence induces between indecomposable objects in \(\mathrm{rep}(Q, \overline{\bmod}\Lambda)\) and noninjective indecomposable objects in \(\mathrm{mono}(Q, \Lambda)\), and we show that the generalized Mimoconstruction, an explicit minimal right approximation into \(\mathrm{mono}(Q, \Lambda)\), gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radial square zero Nakayama algebras, and to give a bijection between the indecomposable 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 bmodulations. In particular, we obtain new results on the singularity category of the algebras \(H\) 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 \(\imath\)quivers algebras that were introduced by Lu and Wang to realize \(\imath\)quantum groups via semiderived Hall algebras.
© 2024 The Author(s). Proceedings of the London Mathematical Society is copyright © London Mathematical Society. is copyright © London Mathematical Society.

MSC:

16G60	Representation type (finite, tame, wild, etc.) of associative algebras
16G20	Representations of quivers and partially ordered sets
13C14	Cohen-Macaulay modules
18A25	Functor categories, comma categories
18C20	Eilenberg-Moore and Kleisli constructions for monads

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