Recollements of Module Categories

TL;DR

This paper explores the structure of module categories through recollements, establishing correspondences with TTF-triples and idempotent ideals, and characterizing when such recollements are induced by idempotent elements.

Contribution

It provides new characterizations of recollements in module categories, linking them to TTF-triples and idempotent ideals, and answers a question about their induced forms.

Findings

Recollements correspond to TTF-triples in abelian categories.

Recollements of module categories relate to idempotent ideals.

Recollements induced by idempotent elements are characterized.

Abstract

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.