Derived Recollements and Generalised AR Formulas

TL;DR

This paper unifies various recollement constructions using derived functors, introduces a duality satisfying generalized Auslander-Reiten formulas, and connects these to classical formulas in representation theory.

Contribution

It presents a general framework for derived recollements and extends Auslander-Reiten formulas through a new duality in stable module theory.

Findings

Derived functors unify different recollement constructions.

The functor W_2 is right exact and induces a duality of defect zero functors.

Generalized Auslander-Reiten formulas encompass classical results.

Abstract

The Defect Recollement, Restriction Recollement, Auslander-Gruson-Jensen Recollement, and others, are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors $\textsf{W}_k:=R_k(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )^*$ are computed and it is shown that the functor $\textsf{W}_2:=R_2(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )^*$ is right exact and restricts to a duality $\textsf{W}$ of the defect zero functors. The duality $\textsf{W}$ satisfies two identities which we call the Generalised Auslander-Reiten formulas. We show that $\textsf{W}$ restricts to the generalised Auslander-Bridger transpose and show that the Generalised Auslander-Reiten formulas reduce to the well-known Auslander-Reiten formulas.