Geodesic
Notes on Several Orthogonal Classes of Flat and $FP$-injective Functors
Matematičeskie zametki, Tome 95 (2014) no. 1, pp. 93-108.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we develop relative homological algebra in the category of functors from finitely presented modules to Abelian groups. More specifically, we introduce the concepts of $\mathfrak{F}$-injective, $\mathfrak{F}$-projective and $\mathfrak{F}$-flat functors. Such functors appear when we study covers and envelopes of functors. The relationships among these functors are investigated and some applications are given.
Keywords: $\mathfrak{F}$-injective functor, $\mathfrak{F}$-projective functor, $\mathfrak{F}$-flat functor, $FP$-injective functor, flat functor, (pre)envelope, (pre)cover. 
@article{MZM_2014_95_1_a7,
     author = {Lixin Mao},
     title = {Notes on {Several} {Orthogonal} {Classes} of {Flat} and $FP$-injective {Functors}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {93--108},
     publisher = {mathdoc},
     volume = {95},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_1_a7/}
}
TY  - JOUR
AU  - Lixin Mao
TI  - Notes on Several Orthogonal Classes of Flat and $FP$-injective Functors
JO  - Matematičeskie zametki
PY  - 2014
SP  - 93
EP  - 108
VL  - 95
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_95_1_a7/
LA  - ru
ID  - MZM_2014_95_1_a7
ER  - 
%0 Journal Article
%A Lixin Mao
%T Notes on Several Orthogonal Classes of Flat and $FP$-injective Functors
%J Matematičeskie zametki
%D 2014
%P 93-108
%V 95
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_95_1_a7/
%G ru
%F MZM_2014_95_1_a7
Lixin Mao. Notes on Several Orthogonal Classes of Flat and $FP$-injective Functors. Matematičeskie zametki, Tome 95 (2014) no. 1, pp. 93-108. http://geodesic.mathdoc.fr/item/MZM_2014_95_1_a7/

[1] M. Auslander, “Coherent Functors”, Proceedings of the Conference on Categorical Algebra (La Jolla, CA, 1965), Springer, New York, 1966, 189–231 | MR | Zbl

[2] M. Auslander, Representation Dimension of Artin Algebras, Queen Mary Coll. Math. Notes, Queen Mary Coll., London, 1971

[3] M. Auslander, I. Reiten, “Stable equivalence of Artin algebras”, Proceedings of the Conference on Orders, Group Rings and Related Topics (Columbus, OH, 1972), Lecture Notes in Math., 353, Springer-Verlag, Berlin, 1973, 8–71 | DOI | MR | Zbl

[4] H. Brune, “On the global dimension of the functor category $((\operatorname{mod} R)^{\mathrm{op}},\mathrm{Ab})$ and a theorem of Kulikov”, J. Pure Appl. Algebra, 28:1 (1983), 31–39 | DOI | MR | Zbl

[5] W. Crawley-Boevey, “Modules of finite length over their endomorphism rings”, Representations of Algebras and Related Topics, London Math. Soc. Lecture Note Ser., 168, Cambridge Univ. Press, Cambridge, 1992, 127–184 pp. | MR | Zbl

[6] W. Crawley-Boevey, “Locally finitely presented additive categories”, Comm. Algebra, 22:5 (1994), 1641–1674 | DOI | MR | Zbl

[7] L. Gruson, C. U. Jensen, “Dimensions cohomologique reliées aux foncteurs $\varprojlim^{(i)}$”, Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1980), Lecture Notes in Math., 867, Springer-Verlag, Berlin, 1981, 234–294 | DOI | MR | Zbl

[8] I. Herzog, “The phantom cover of a module”, Adv. Math., 215:1 (2007), 220–249 | DOI | MR | Zbl

[9] I. Herzog, “Contravariant functors on the category of finitely presented modules”, Israel J. Math., 167 (2008), 347–410 | DOI | MR | Zbl

[10] M. Prest, Model Theory and Modules, London Math. Soc. Lecture Note Ser., 130, Cambridge Univ. Press, Cambridge, 1988 | MR | Zbl

[11] M. Saorín, A. Del Valle, “Covers and envelopes in functor categories”, Interactions Between Ring Theory and Representations of Algebras, Lecture Notes in Pure and Appl. Math., 210, Marcel Dekker, New York, 2000, 321–329 pp. | MR | Zbl

[12] B. Stenström, Rings of Quotients. An Introduction to Methods of Ring Theory, Grundlehren Math. Wiss., 217, Springer-Verlag, Berlin, 1975 | MR | Zbl

[13] E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, de Gruyter Exp. Math., 30, Walter de Gruyter, Berlin, 2000 | MR | Zbl

[14] E. E. Enochs, “Injective and flat covers, envelopes and resolvents”, Israel J. Math., 39:3 (1981), 189–209 | DOI | MR | Zbl

[15] E. E. Enochs, L. Oyonarte, Covers, Envelopes and Cotorsion Theories, Nova Sci. Publ., New York, 2002

[16] N. Ding, “On envelopes with the unique mapping property”, Comm. Algebra, 24 (1996), 1459–1470 | DOI | MR | Zbl

[17] R. Göbel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Exp. Math., 41, Walter de Gruyter, Berlin, 2006 | DOI | MR | Zbl

[18] J. J. Rotman, An Introduction to Homological Algebra, Pure Appl. Math., 85, Academic Press, New York, 1979 | MR | Zbl