Ladders of recollements, categories of monomorphisms and singularity categories

TL;DR

This paper explores the structure of derived and singularity categories related to monomorphism categories over Artin algebras, revealing infinite ladders of recollements and establishing connections with Gorenstein properties.

Contribution

It introduces and characterizes infinite ladders of recollements in derived categories of monomorphism and morphism categories, and studies their singularity categories and Gorenstein subcategories.

Findings

Derived categories of monomorphism categories admit infinite ladders of recollements.

A periodic infinite ladder connects the singularity category of the monomorphism category with the standard singularity category.

Conditions are provided for singular equivalences between monomorphism categories of different algebras.

Abstract

In this paper we show that the (un)bounded derived categories$\colon$(i) of the monomorphism category, (ii) of the morphism category and (iii) of the double morphism category, admit a periodic infinite ladder of recollements. These results are based on a characterization that we provide for a recollement of (compactly generated) triangulated categories to admit a ladder of some height going either upwards or downwards. Moreover, we introduce and study the singularity category of the monomorphism category over an Artin algebra $\Lambda$ and show that there is a periodic infinite ladder that connects this triangulated category with the standard singularity category of $\Lambda$. We also provide sufficient conditions for the monomorphism categories of two algebras to be singularly equivalent. The last aim of this paper is to study monomorphisms where the domain has finite projective dimension. We show that the latter category is a Gorenstein subcategory of the monomorphism category when $\Lambda$ is a Gorenstein Artin algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory, and show that it carries a structure of a Gorenstein abelian category.