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 Wk:=Rk(_)⁎ 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.