The Auslander-Gruson-Jensen Recollement

Abstract

For any ring $R$, the Auslander-Gruson-Jensen functor is the exact contravariant functor $${\text{D}}_A:\text{fp}(\text{Mod}(R),\text{Ab})⟶(\text{mod}(R^{op}),\text{Ab})$$ sending representable functors $(X,\underline{})$ to tensor functors $X\otimes \underline{}$. We show that this functor admits a fully faithful left adjoint ${\text{D}}_L$ and a fully faithful right adjoint ${\text{D}}_R$. The left adjoint $${\text{D}}_L(\text{mod}(R^{op}),\text{Ab})⟶\text{fp}(\text{Mod}(R),\text{Ab})$$ induces an equivalence of categories $$\frac{\text{fp}(\text{Mod}(R),\text{Ab})}{\{F|{\text{D}}_{A}F=0\}}\cong (\text{mod}(R^{op}),\text{Ab}{)}^{op}$$ where $\{F|{\text{D}}_{A}F=0\}$ is the Serre subcategory of $\text{fp}(\text{Mod}(R),\text{Ab})$ consisting of all functors $F$ arising from pure exact sequences. As a result, the functor ${\text{D}}_A$ is seen to be a Serre localization functor. The right adjoint $${\text{D}}_R:(\text{mod}(R^{op}),\text{Ab})⟶\text{fp}(\text{Mod}(R),\text{Ab})$$ together with ${\text{D}}_A$ restricts to the well known Auslander-Gruson-Jensen duality.

11 pages. Updating to match published version