The Auslander–Gruson–Jensen recollement

Abstract

For any ring R, the Auslander–Gruson–Jensen functorDA: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 DR and a fully faithful left adjoint DL. That is, we show that DA is part of a recollement of abelian categories. In particular, this shows that DA is a localisation and a colocalisation which gives an equivalence of categoriesfp(R-Mod,Ab){F:DAF=0}≃(mod-R,Ab)op. We show that {F:DAF=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.