Derived Recollements and Generalised AR Formulas

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 ${\text{W}}_k:=R_k(\underline{}{)}^{\ast }$ are computed and it is shown that the functor ${\text{W}}_2:=R_2(\underline{}{)}^{\ast }$ is right exact and restricts to a duality $\text{W}$ of the defect zero functors. The duality $\text{W}$ satisfies two identities which we call the Generalised Auslander-Reiten formulas. We show that $\text{W}$ restricts to the generalised Auslander-Bridger transpose and show that the Generalised Auslander-Reiten formulas reduce to the well-known Auslander-Reiten formulas.