Approximations and cotorsion pairs related to a tilting pair

Abstract

The notion of a tilting pair over artin algebras was introduced by Miyashita in 2001. It is a useful tool in the tilting theory. Approximations and cotorsion pairs related to a fixed tilting pair were discussed. A contravariantly (covariantly) finite subcategory and a cotorsion pair associated with a fixed tilting pair were given in this paper.

Keywords

Selforthogonal module / contravariantly (covariantly) finite subcategory / cotorsion pair / tilting pair

References

[1] Anderson F D, Fuller K R. Rings and Categories of Modules, 1992 2nd ed. Berlin, New York: Springer-Verlag

[2] Araya T, Yoshino Y. Remark on a depth formula, a grade inequality, and a conjecture of Auslander. Comm Algebra, 1998, 26: 3793-3806

[3] Auslander M, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86: 111-152

[4] Auslander M, Smalø S O. Preprojective modules over artin algebras. J Algebra, 1980, 66: 61-122

[5] Avramov L L, Buchweitz R O. Support varieties and cohomology over complete intersections. Invent Math, 2000, 142: 285-318

[6] Bazzoni S. A characterization of n-cotilting and n-tilting modules. J Algebra, 2004, 273: 359-372

[7] Enochs E E, Jenda O M G. Relative Homological Algebra, 2000, Berlin, New York: Walter de Gruyter

[8] Enochs E E, Oyonarte L. Covers, Envolopes, and Cotorsion Theories, 2002, New York: Nova Science Publishers, Inc

[9] Garcia R J R. Covers and Envelopes in the Category of Complexes of Modules, 1999, London: Chapman & Hall/CRC

[10] G:obel R, Trlifaj J. Approximations and Endomorphism Algebras of Modules, 2006, Berlin, New York: Walter de Gruyter

[11] Miyashita Y. Tilting modules associated with a series of idempotent ideals. J Algebra, 2001, 238: 485-501

[12] Reiten I. Tilting theory and homologically finite subcategories with applications to quasihereditary algebras. Handbook of Tilting Theory. London Math Soc Lect Note Ser, 332, 2007, Cambridge: Cambridge Univ Press 179 214

[13] Sega L M. Vanishing of cohomology over Gorenstein rings of small codimension. Proc Amer Math Soc, 2002, 131: 2313-2323

[14] Trlifaj J. Infinite dimensional tilting modules and cotorsion pairs. Handbook of Tilting Theory. London Math Soc Lect Note Ser, 332, 2007, Cambridge: Cambridge Univ Press 279 322

[15] Wei J Q, Xi C C. A characterization of the tilting pair. J Algebra, 2007, 317(1): 376-391