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Finiteness criteria in quasi-resolving subcategories. (English) Zbl 1451.18025

The main purpose of the paper under review is to generalize some results by I. Emmanouil and O. Talelli [Trans. Am. Math. Soc. 366, No. 12, 6329–6351 (2014; Zbl 1354.16012)] to the setting of quasi-resolving subcategories of an abelian category given by X. Zhu [J. Algebra 414, 6–40 (2014; Zbl 1386.18048)]. Then the authors apply their results to some known categories such as the category of Gorenstein projective modules/complexes and the category of Gorenstein AC-projective modules/complexes.
Reviewer: Li Liang (Lanzhou)

MSC:

18G20 Homological dimension (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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[1] Auslander, M., Bridger, M.: Stable Module Theory, No. 94. American Mathematical Society, Providence, Rhode Island (1969) · Zbl 0204.36402 · doi:10.1090/memo/0094
[2] Avramov, L.L., Martsinkovsky, A.: Absolute, relative and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc 85, 393-440 (2002) · Zbl 1047.16002 · doi:10.1112/S0024611502013527
[3] Bennis, D., Ouarghi, \[K.: {\cal{X}}\] X-Gorenstein projective modules. Int. Math. Forum 5, 487-491 (2010) · Zbl 1191.13014
[4] Bravo, D., Gillespie, J.: Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44, 2213-2233 (2016) · Zbl 1346.18021 · doi:10.1080/00927872.2015.1044100
[5] Bravo, D., Gillespie, J., Hovey, M.: The stable module category of a general ring (2014). arXiv:1405.5768v1
[6] Christensen, L.W.: Gorenstein Dimensions. Lecture Notes in Mathematics, vol. 1747. Springer, Berlin (2000) · Zbl 0965.13010 · doi:10.1007/BFb0103980
[7] 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) · Zbl 1104.13008 · doi:10.1016/j.jalgebra.2005.12.007
[8] Ding, N.Q., Li, Y.L., Mao, L.X.: Strongly Gorenstein flat modules. J. Aust. Math. Soc. 86, 323-338 (2009) · Zbl 1200.16010 · doi:10.1017/S1446788708000761
[9] Emmanouil, I., Talelli, O.: Finiteness criteria in Gorenstein homological algebra. Trans. Am. Math. Soc 366, 6329-6351 (2014) · Zbl 1354.16012 · doi:10.1090/S0002-9947-2014-06007-8
[10] Enochs, E.E., Garcí Rozas, J.R.: Gorenstein injective and projective complexes. Commun. Algebra 5, 1657-1674 (1998) · Zbl 0908.18007 · doi:10.1080/00927879808826229
[11] Enochs, E.E., Jenda, O.M.G.: Gorenstein injective and projective modules. Math. Z. 220, 611-633 (1995) · Zbl 0845.16005 · doi:10.1007/BF02572634
[12] Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189, 167-193 (2004) · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007
[13] Sather-Wagstaff, S., Sharif, T., White, D.: Stability of Gorenstein categories. J. Lond. Math. Soc. 77, 481-502 (2008) · Zbl 1140.18010 · doi:10.1112/jlms/jdm124
[14] Yang, G., Liu, Z.K., Liang, L.: Ding projective and Ding injective modules. Algebra Colloq. 20, 601-612 (2013) · Zbl 1280.16003 · doi:10.1142/S1005386713000576
[15] Yang, G., Liu, Z.K., Liang, L.: Model structures on categories of complexes over Ding-Chen rings. Commun. Algebra 41, 50-69 (2013) · Zbl 1273.55009 · doi:10.1080/00927872.2011.622326
[16] Yang, X.Y., Liu, Z.K.: Gorenstein projective, injective and flat complexes. Commun. Algebra 5, 1705-1721 (2011) · Zbl 1238.16002 · doi:10.1080/00927871003741497
[17] Zhao, R.Y., Ding, N.Q.: \[ \cal{(W, Y, X)}(W,Y\],X)-Gorenstein complexes. Commun. Algebra 45, 3075-3090 (2017) · Zbl 1375.18072 · doi:10.1080/00927872.2016.1235173
[18] Zhu, X.S.: Resolving resolution dimensions. Algebra Represent. Theory 16, 1165-1191 (2013) · Zbl 1296.18014 · doi:10.1007/s10468-012-9351-5
[19] Zhu, X.S.: The homological theory of quasi-resolving subcategories. J. Algebra 414, 6-40 (2014) · Zbl 1386.18048 · doi:10.1016/j.jalgebra.2014.05.018
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