Equivalences Induced by Adjoint Functors

Abstract

Let 𝒜 and ℬ be two Grothendieck categories, R : 𝒜 → ℬ, L : ℬ → 𝒜 a pair of adjoint functors, S ∈ ℬ a generator, and U = L(S). U defines a hereditary torsion class in 𝒜, which is carried by L, under suitable hypotheses, into a hereditary torsion class in ℬ. We investigate necessary and sufficient conditions which assure that the functors R and L induce equivalences between the quotient categories of 𝒜 and ℬ modulo these torsion classes. Applications to generalized module categories, rings with local units and group graded rings are also given here.

Key Words:

• Adjoint pair
• Equivalence
• Localization
• Grothendieck categories
• Small preadditive category

1991 Mathematics Subject Classification:

• 18E15
• 18E35
• 18E40
• 16D90
• 18A40

Acknowledgments

This paper was written while the author was visiting the Mathematical Institute of the “Friedrich Schiller” University, Jena. He is grateful to the Alexander von Humboldt Foundation for the financial support, and to Professor Burkhard Külshammer for his help and hospitality. The author would like to thank to Andrei Marcus for many discussions and comments. In addition, the author is grateful to an anonymous referee for a number of suggestions.