A Characterization of n-Gorenstein Tilting Comodules
Tóm tắt
The aim of this paper is to introduce the concept of n-Gorenstein tilting comodules and study its main properties. This concept generalizes the notion of n-tilting comodules of finite injective dimensions to the case of finite Gorenstein injective dimensions. As an application of our results, we discuss the problem of existence of complements to partial n-Gorenstein tilting comodules.
Tài liệu tham khảo
Asensio, M.J., López-Ramos, J.A., Torrecillas, B.: Gorenstein coalgebras. Acta Math. Hungar 85(1), 187–198 (1999)
Auslander, M., Smalø, S.: Preprojective modules over artin algebras. J. Algebra 66, 61–122 (1980)
Auslander, M., Solberg, Ø.: Relative homology and representation theory I, relative homology and homologically finite subcategories. Comm. Algebra 21(9), 2995–3031 (1993)
Auslander, M., Solberg, Ø.: Relative homology and representation theory II, relative cotilting theory. Comm. Algebra 21(9), 3033–3079 (1993)
Auslander, M., Solberg, Ø.: Relative homology and representation theory III, cotilting modules and Wedderburn correspondence. Comm. Algebra 21(9), 3081–3097 (1993)
Bazzoni, S.: A characterization of n-cotilting and n-tilting modules. J. Algebra 273(1), 359–372 (2004)
Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. Rrpresentation Theory II. Springer, Berlin Heidelberg (1980)
Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178(1–2), 614–634 (1995)
D\(\breve{\text{a}}\)sc\(\breve{\text{ a }}\)lescu, S. N\(\breve{\text{ a }}\)st\(\breve{\text{ a }}\)sescu, C. Raianu, S.: Hopf Algebras. An Introduction, Lecture Notes in Pure and Applied Mathematics, No. 235, Marcel-Dekker, New-York, 2001
Eilenberg, S., Moore, J.C.: Foundations of relative homological algebra. American Mathematical Society, (1965)
Enochs, E.E., Jenda, O.M.G.: Relative homological algebras. de Gruyter Expositions in Mathematics, Vol. 30, Walter de Gruyter, Berlin-New York, (2000)
Enochs, E.E., López-Ramos, J.A.: Relative homological coalgebras. Acta Math. Hungar 104(4), 331–343 (2004)
Fu, X.R., Yao, H.L.: The Auslander-Reiten formula for comodule categories with applications to partial tilting comodules and tilting global dimension. Colloq Math. 153(2), 219–240 (2018)
García Rozas, J.R., López Ramos, J.A., Torrecillas, B.: Semidualizing and tilting Adjoint Pairs, applications to comodules. Bull. Malays. Math. Sci. Soc. 38(1), 197–218 (2015)
Gómez-Torrecillas, J., Năstăsescu, C.: Quasi-co-Frobenius coalgebras. J. Algebra 174(3), 909–923 (1995)
Happel, D., Ringel, C.M.: Tilted algebras. Trans. Am. Math. Soc. 274(2), 399–443 (1982)
Hügel, L.A., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13(2), 239–250 (2001)
Hügel, L.A., Tonolo, A., Trlifaj, J.: Tilting preenvelopes and cotilting precovers. Algebras Represent Theory 4(2), 155–170 (2001)
Li, Y., Yao, H.L.: Tilting torsion-free classes in the category of comodules. Bull. Iran. Math. Soc. 47(2), 553–567 (2021)
Li, H.H., Wang, J.F., Huang, Z.Y.: Applications of balanced pairs. Sci. China Math. 59(5), 861–874 (2016)
Lin, L.P.: Semiperfect coalgebras. J. Algebra. 49(2), 357–373 (1977)
Martínez, L., Mendoza, O.: Relative torsion classes, relative tilting, and relative silting modules. Comm. Algebra. 50(5), 1858–1883 (2022)
Meng, F.Y.: (Weakly) Gorenstein injective and (Weakly) Gorenstein coflat comodules. Studia Sci. Math. Hungar 1(49), 106–119 (2012)
Miyashita, Y.: Tilting modules of finite projective dimension. Math. Z. 193(1), 113–146 (1986)
Pan, Q.X., Li, Q.: On Gorenstein injective and injective comodules. Math. Notes. 2(94), 255–265 (2013)
Rada, J., Saorin, M.: Rings characterized by (pre)envelopes and (pre)covers of their modules. Comm. Algebra 26, 899–912 (1998)
Simson, D.: Coalgebras, comodules, pseudocompact algebras and tame comodule type. Colloq. Math. 90, 101–150 (2001)
Simson, D.: Cotilted coalgebras and tame comodule type. Arab. J. Sci. Eng. Sect. C. Theme Issues 33(2), 421–445 (2008)
Takeuchi, M., Iwahori, N.: Morita theorems for categories of comodules. J. Math. Sci. Univ. Tokyo. Sect A Math. 24(3), 629–644 (1977)
Wang, M.Y.: Some co-hom functors and classical tilting comodules. Southeast Asian Bull. Math. 22(4), 455–468 (1998)
Wang, M.Y.: Tilting comodules over semi-perfect coalgebras. Algebra Colloq. 6(4), 461–472 (1999)
Wei, J.Q.: A note on relative tilting modules. J. Pure Appl. Algebra 214(4), 493–500 (2010)
Yan, L., Li, W., Ouyang, B.: Gorenstein cotilting and tilting modules. Comm. Algebra 44(2), 591–603 (2016)