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 zeroth derived functors and methods from stable module theory. The right derived functors W_k:=R_k(_)^⁎ are computed and it is shown that the functor W₂:=R₂(_)^⁎ is right exact and restricts to a duality W of the defect zero functors. The duality W satisfies two identities which we call the Generalised Auslander–Reiten formulas. We show that W induces the generalised Auslander–Bridger transpose and show that the Generalised Auslander–Reiten formulas reduce to the well-known Auslander–Reiten formulas.