Revising Auslander-Gruson-Jensen duality

TL;DR

This paper revises and clarifies the Auslander-Gruson-Jensen duality by providing a simpler description of the free abelian category, enhancing understanding of dualities between definable subcategories of modules.

Contribution

It offers a straightforward description of the free abelian category, simplifying the understanding of Auslander-Gruson-Jensen duality and related module category dualities.

Findings

Clarifies the structure of the free abelian category.

Simplifies the understanding of Auslander-Gruson-Jensen duality.

Enhances the conceptual framework for module category dualities.

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 $\rm mod\text{-}(mod\text{-}A)\rightarrow mod\text{-}(mod\text{-}A^{op})$. The existence of this duality arises from the fact that $\rm mod\text{-}(mod\text{-}A)$ 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.