Derived Recollements and Generalised AR Formulas

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.