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 2 B a generator, and U ¼ 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.

Localizations, colocalizations and non additive ∗-objects

“…These results may be also compared with [9,Theorem 1.18], where the framework is also that of abelian categories.…”

“…The following are equivalent: , where the work is done in the setting of abelian categories, and the proof stresses the abelian structure. These results may be also compared with [9,Theorem 1.18], where the framework is also that of abelian categories.…”

Generalized lax epimorphisms in the additive case

“…In this sense it is an additive version of [2, Theorem 4.1] (see also [8,Corollary 4.5]). But it also gives a partial answer to a question occurring naturally in [12]: Given two Grothendieck categories, A and B, a pair of adjoint functors between them R : A → B at the right and L : B → A at the left, and a hereditary torsion class T in A, what additional hypotheses should be considered, such that {B ∈ B | LB ∈ T } is a hereditary torsion class?…”