Homological dimensions of complexes related to cotorsion pairs

Let (A, B) be a cotorsion pair in R-Mod. We define and study notions of A dimension and B dimension of unbounded complexes, which is given by means of dg-projective resolution and dg-injective resolution, respectively. As an application, we extend the Gorenstein flat dimension of complexes, which was defined by Iacob. Gorenstein cotorsion, FP-projective, FP-injective, Ding projective, and Ding injective dimension are also extended from modules to complexes. Moreover, we characterize Noetherian rings, von Neumann regular rings, and QF rings by the FP-projective, FP-injective, and Ding projective (injective) dimension of complexes, respectively.

Anahtar Kelimeler:

Cotorsion pairs, A dimension of complexes, B dimension of complexes

Homological dimensions of complexes related to cotorsion pairs

Keywords:

Cotorsion pairs, A dimension of complexes, B dimension of complexes,

PDF

___

Asadollahi, J., Salarian, S.: Gorenstein injective dimension for complexes and Iwanaga-Gorenstein rings. Comm. Algebra 34, 3009–3022 (2006).
Avramov, L., Foxby, H.B.: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71, 129–155 (1991).
Bennis, D.: Rings over which the class of Gorenstein flat modules is closed under extensions. Comm. Algebra 37, 855–868 (2009).
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton, NJ, USA. Princeton University Press, 1999 reprint of 1956 original.
Christensen, L.W.: Gorenstein Dimensions. Lecture Notes in Mathematics Vol. 1747. Berlin. Springer-Verlag 2000.
Christensen, L.W., Frankild, A., Holm, H.: On Gorenstein projective, injective and flat dimensions-a functorial description with applications. J. Algebra 302, 231–279 (2006).
Ding N.Q., Chen J.L.: The flat dimensions of injective modules. Manuscripta Math. 78, 165–177 (1993).
Ding N.Q., Chen J.L.: Coherent rings with finite self-FP-injective dimension. Comm. Algebra 24, 2963–2980 (1996).
Ding N.Q., Mao L.X.: Relative FP-projective modules. Comm. Algebra 33, 1587–1602 (2005).
Ding N.Q., Li Y.L., Mao L.X.: Strongly Gorenstein flat modules. J. Aust. Math. Soc. 86, 323–338 (2009).
Enochs, E.E.: Injective and flat covers, envelopes and resolvents. Israel J. Mathematics 39, 189–209 (1981).
Enochs, E.E., Jenda, O.M.G., Torrecillas, B.: Gorenstein flat modules. Nanjing Daxue Xuebao Shuxue Bannian Kan 10, 1–9 (1993).
Enochs, E.E., Jenda, O.M.G., Xu, J.: Orthogonality in the category of complexes. Math. J. Okayama Univ. 38, 25–46 (1996).
Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra. de Gruyter Exp. Math., Vol. 30. Berlin. de Gruyter 2000.
Enochs, E.E., Oyonarte, L.: Covers, Envelopes and Cotorsion Theories. Hauppauge, NY, USA. Nova Science Publishers 2002.
Enochs, E.E., Estrada, S., Torrecillas, B.: Gorenstein flat covers and Gorenstein cotorsion modules over integral group rings. Algebr. Represent. Theory 8, 525–539 (2005).
Garc´ıa Rozas, J.R.: Covers and Envelopes in the Category of Complexes of Modules. Boca Raton, FL, USA. CRC Press 1999.
Gillespie, J.: The flat model structure on Ch( R ). Tran. Amer. Math. Soc. 356, 3369–3390 (2004).
Gillespie, J.: Model structures on modules over Ding-Chen rings. Homology, Homotopy Appl. 12, 61–73 (2010).
Iacob, A.: Gorenstein flat dimension for complexes. J. Math. Kyoto Univ. 49, 817–842 (2009).
Mao, L.X., Ding, N.Q.: FP-projective dimension. Comm. Algebra 33, 1153–1170 (2005).
Mao, L.X., Ding, N.Q.: The cotorsion dimension of modules and rings. Abelian Groups, Rings, Modules, and Homological Algebra, Lect. Notes Pure Appl. Math. 249, 217–233 (2006).
Mao, L.X., Ding, N.Q.: Envelopes and covers by modules of finite FP-injective and flat dimensions. Comm. Algebra 35, 833–849 (2007).
Mao, L.X., Ding N.Q.: Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7, 491–506 (2008).
Salce, L.: Cotorsion theories for abelian groups. Symposia Math. 23, (1979).
Stenstr¨om, B.: Coherent rings and FP-injective modules. J. London Math. Soc. 2, 323–329 (1970).
Veliche, O.: Gorenstein projective dimension for complexes. Trans. Amer. Math. Soc. 358, 1257–1283 (2006).
Xu, J.Z.: Flat Covers of Modules. Lecture Notes in Mathematics Vol. 1634. Berlin. Springer-Verlag 1996.
Yang, G., Liu, Z.K.: Cotorsion pairs and model structure on Ch(R). Proc. Edinb. Math. Soc. 54, 783–797 (2011).
Yang, G., Liu, Z.K.: Gorenstein flat covers over GF-closed rings. Comm. Algebra 40, 1632–1640 (2012).
Yassemi, S.: Gorenstein dimension. Math. Scand. 77, 161–174 (1995).