Ladders of recollements of abelian categories

TL;DR

This paper introduces ladders of recollements in abelian categories, enabling the construction of triangulated category recollements, tilting of abelian recollements, and preservation of properties like Gorenstein conditions.

Contribution

It develops the theory of ladders of recollements, providing new methods to relate abelian and triangulated categories and to preserve important properties.

Findings

Ladders of a certain height construct triangulated recollements from abelian ones.

Ladders facilitate tilting of abelian recollements.

Ladders ensure preservation of Gorenstein properties in functors.

Abstract

Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.