Injective stabilization of additive functors. II. (Co)torsion and the Auslander-Gruson-Jensen functor

JOURNAL OF ALGEBRA

卷:548

Article

Martsinkovsky, Alex¹  Russell, Jeremy² 

[1] Northeastern Univ, Dept Math, Boston, MA 02115 USA

[2] Rowan Univ, Dept Math, 201 Mull Hill Rd, Glassboro, NJ 08028 USA

关键词: Additive functor;    Injective stabilization;    Projective stabilization;    Classical torsion;    Bass torsion functor;    Injective torsion;    Torsion submodule of a module;    Cotorsion quotient module of a module;    Torsion module;    Torsion-free module;    Cotorsion module;    Cotorsion-free module;    Auslander-Gruson-Jensen functor;    Filtered colimits;    Finitely presented functor;    Reject of fiats;    Trace of injectives;    Cosatellite;    pp-functors;    Pure injective;    Absolutely pure module;    Artin algebra;    Character module;   

DOI  :  10.1016/j.jalgebra.2019.11.016

来源: Elsevier

【 摘 要 】

The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach - based on projective stabilization - gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality - the new concept is defined for any module over any ring. Unlike torsion, cotorsion does not have classical prototypes. General properties of these constructs are established. It is shown that the Auslander-Gruson-Jensen functor applied to the cotorsion functor returns the torsion functor. As a consequence, a ring is one-sided absolutely pure if and only if each pure injective on the other side is cotorsion-free. If the injective envelope of the ring is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor applied to the torsion functor returns the cotorsion functor. This correspondence establishes a duality between torsion and cotorsion over such rings. In particular, this duality applies to artin algebras. It is also shown that, over any ring, the character module of the torsion of a module is isomorphic to the cotorsion of the character module of the module. Under various finiteness conditions on the injective envelope of the ring, the derived functors of torsion and cotorsion are computed. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free