On a Natural Duality Between Grothendieck Categories

Abstract

A right R-module M with endomorphism ring S is called a costar module if it induces the duality

between the class of M _R -torsionless right R-modules X with Hom _R (X, M _R ) finitely generated over S and the class of _S M-torsionless finitely generated left S-modules. In this article we consider, more generally, a pair of additive, contravariant functors—which are adjoint on the right—between Grothendieck categories, and describe a natural duality induced by them. Our result subsumes the situation mentioned above but also, for example, a rigid graded duality that gives the notion of graded costar module.

Key Words:

• Duality
• Graded module
• Reflexive module

2000 Mathematics Subject Classification:

• 16E30
• 16D90

ACKNOWLEDGMENT

The author would like to thank the referee for his comments and suggestions that improved the article.

Research partially supported by Spanish Project (MTM2005–03227) from MCT.

Notes

Communicated by J. L. Gomez Pardo.