The Auslander–Gruson–Jensen recollement

Abstract For any ring R , the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.