Revising Auslander-Gruson-Jensen duality

Abstract

For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\mathrm{mod}\text{-}(\mathrm{mod}\text{-}\mathrm{A})\to \mathrm{mod}\text{-}(\mathrm{mod}\text{-}{\mathrm{A}}^{\mathrm{op}})$. The existence of this duality arises from the fact that $\mathrm{mod}\text{-}(\mathrm{mod}\text{-}\mathrm{A})$ is the free abelian category over the pre-additive category $A$ with a single object. In this note, first, we give a simple description of the free abelian category. This description clarifies Auslender-Gruson-Jensen duality and also the duality between definable subcategories of right $A$-modules and those of left $A$-modules.

Major revision