Auslander-Reiten duality for Grothendieck abelian categories

TL;DR

This paper extends Auslander-Reiten duality to Grothendieck abelian categories with finitely presented objects, revealing new duality properties and connections to Serre duality in algebraic geometry.

Contribution

It generalizes Auslander-Reiten duality to a broader class of categories and links it to Serre duality for schemes, providing new insights into duality theories.

Findings

Auslander-Reiten duality is extended to Grothendieck abelian categories.

The functor Ext^1(C,-) admits a partially defined right adjoint for finitely presented C.

Connections between Auslander-Reiten duality and Serre duality are discussed.

Abstract

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.