Properties of abelian categories via recollements

TL;DR

This paper studies how key properties of abelian categories are preserved or transferred through recollements, revealing that certain properties like being Grothendieck are maintained, but the converse may not always hold.

Contribution

It investigates the behavior of fundamental properties of abelian categories under recollements, providing conditions for property transfer and counterexamples.

Findings

Recollements preserve Grothendieck abelian properties.

Counterexample shows the converse of property transfer does not always hold.

Multiple conditions identified for property preservation in recollements.

Abstract

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.