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Comptes Rendus. Mathématique

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Relative global dimensions and stable homotopy categories
Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 379-392.

In this paper we study the finiteness of global Gorenstein AC-homological dimensions for rings, and answer the questions posed by Becerril, Mendoza, Pérez and Santiago. As an application, we show that any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category, which improves the known result by Beligiannis.

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DOI : https://doi.org/10.5802/crmath.50
Classification : 18G25,  18G20
@article{CRMATH_2020__358_3_379_0,
     author = {Li Liang and Junpeng Wang},
     title = {Relative global dimensions and stable homotopy categories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {379--392},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.50},
     language = {en},
}
Li Liang; Junpeng Wang. Relative global dimensions and stable homotopy categories. Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 379-392. doi : 10.5802/crmath.50. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.50/

[1] Maurice Auslander; Mark Bridger Stable module theory, Memoirs of the American Mathematical Society, 94, American Mathematical Society, 1969 | MR 0269685 | Zbl 0204.36402

[2] Víctor Becerril; Octavio Mendoza; Marco A. Pérez; Valente Santiago Frobenius pairs in abelian categories. Correspondences with cotorsion pairs, exact model categories, and Auslander–Buchweitz contexts, J. Homotopy Relat. Struct., Volume 14 (2019) no. 1, pp. 1-50 | Article | MR 3913970 | Zbl 07055725

[3] Apostolos Beligiannis The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co-)stabilization, Commun. Algebra, Volume 28 (2000) no. 10, pp. 4547-4596 | Article | MR 1780017 | Zbl 0964.18006

[4] Apostolos Beligiannis Homotopy theory of modules and Gorenstein rings, Math. Scand., Volume 89 (2001) no. 1, pp. 5-45 | Article | MR 1856980 | Zbl 1023.55009

[5] Apostolos Beligiannis; Idun Reiten Homological and homotopical aspects of torsion theories, Memoirs of the American Mathematical Society, 883, American Mathematical Society, 2007, viii+207 pages | MR 2327478 | Zbl 1124.18005

[6] Driss Bennis Rings over which the class of Gorenstein flat modules is closed under extensions, Commun. Algebra, Volume 37 (2009) no. 3, pp. 855-868 | Article | MR 2503181 | Zbl 1205.16006

[7] Driss Bennis; Najib Mahdou Global Gorenstein dimensions, Proc. Am. Math. Soc., Volume 138 (2010) no. 2, pp. 461-465 | Article | MR 2557164 | Zbl 1205.16007

[8] Daniel Bravo; Sergio Estrada; Alina Iacob FP n -injective, FP n -flat covers and preenvelopes, and Gorenstein AC-flat covers, Algebra Colloq., Volume 25 (2018) no. 2, pp. 319-334 | Article | MR 3805326 | Zbl 1427.18007

[9] Daniel Bravo; James Gillespie; Mark Hovey The stable module category of a general ring (2014) (https://arxiv.org/abs/1405.5768)

[10] Ragnar-Olaf Buchweitz Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, 1986 University of Hannover (Germany), http://hdl.handle.net/1807/16682

[11] Xiao-Wu Chen Relative singularity categories and Gorenstein-projective modules, Math. Nachr., Volume 284 (2011) no. 2-3, pp. 199-212 | Article | MR 2790881 | Zbl 1244.18014

[12] Lars Winther Christensen; Sergio Estrada; Peder Thompson Homotopy categories of totally acyclic complexes with applications to the flat-cotorsion theory (2019) (https://arxiv.org/abs/1812.04402v2, to appear in Contemporary Mathematics)

[13] Zhenxing Di; Zhongkui Liu; Xiaoyan Yang; Xiaoxiang Zhang Triangulated equivalence between a homotopy category and a triangulated quotient category, J. Algebra, Volume 506 (2018), pp. 297-321 | MR 3800079 | Zbl 1391.18017

[14] Ioannis Emmanouil On the finiteness of Gorenstein homological dimensions, J. Algebra, Volume 372 (2012), pp. 376-396 | Article | MR 2990016 | Zbl 1286.13014

[15] Edgar E. Enochs; Overtoun M. G. Jenda Gorenstein injective and projective modules, Math. Z., Volume 220 (1995) no. 4, pp. 611-633 | Article | MR 1363858 | Zbl 0845.16005

[16] Edgar E. Enochs; Overtoun M. G. Jenda; Blas Torrecillas Gorenstein flat modules, J. Nanjing Univ., Math. Biq., Volume 10 (1993) no. 1, pp. 1-9 | MR 1248299 | Zbl 0794.16001

[17] Edgar E. Enochs; Overtoun M. G. Jenda; Jinzhong Xu Orthogonality in the category of complexes, Math. J. Okayama Univ., Volume 38 (1996), pp. 25-46 | MR 1644453 | Zbl 0940.18006

[18] Sergio Estrada; James Gillespie The projective stable category of a coherent scheme, Proc. R. Soc. Edinb., Sect. A, Math., Volume 149 (2019) no. 1, pp. 15-43 | Article | MR 3922806 | Zbl 1423.18053

[19] Sergio Estrada; Alina Iacob; Marco A. Pérez Model structures and relative Gorenstein flat modules and chain complexes (2018) (https://arxiv.org/abs/1709.00658v2, to appear in Contemporary Mathematics)

[20] James Gillespie The flat model structure on Ch (R), Trans. Am. Math. Soc., Volume 356 (2004) no. 8, pp. 3369-3390 | Article | MR 2052954 | Zbl 1056.55011

[21] James Gillespie Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., Volume 291 (2016), pp. 859-911 | Article | MR 3459032 | Zbl 1343.18013

[22] James Gillespie Hereditary abelian model categories, Bull. Lond. Math. Soc., Volume 48 (2016) no. 6, pp. 895-922 | Article | MR 3608936 | Zbl 1372.18001

[23] James Gillespie On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mt. J. Math., Volume 47 (2017) no. 8, pp. 2641-2673 | Article | MR 3760311 | Zbl 06840993

[24] James Gillespie On the homotopy category of AC-injective complexes, Front. Math. China, Volume 12 (2017) no. 1, pp. 97-115 | Article | MR 3579262 | Zbl 1397.18035

[25] James Gillespie AC-Gorenstein rings and their stable module categories, J. Aust. Math. Soc., Volume 107 (2019) no. 2, pp. 181-198 | Article | MR 4001567 | Zbl 07104405

[26] Mark Hovey Cotorsion pairs, model category structures, and representation theory, Math. Z., Volume 241 (2002) no. 3, pp. 553-592 | Article | MR 1938704 | Zbl 1016.55010

[27] Ellen Kirkman; James Kuzmanovich On the global dimension of fibre products, Pac. J. Math., Volume 134 (1988) no. 1, pp. 121-132 | Article | MR 953503 | Zbl 0617.16014

[28] Henning Krause Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math., Volume 139 (2000) no. 1, pp. 99-133 | Article | MR 1728877 | Zbl 0937.18013

[29] Henning Krause The stable derived category of a Noetherian scheme, Compos. Math., Volume 141 (2005) no. 5, pp. 1128-1162 | Article | MR 2157133 | Zbl 1090.18006

[30] Lixin Mao; Nanqing Ding The cotorsion dimension of modules and rings, Abelian groups, rings, modules, and homological algebra (Lecture Notes in Pure and Applied Mathematics), Volume 249, Chapman & Hall/CRC, 2006, pp. 217-233 | MR 2229114 | Zbl 1110.16005

[31] Dmitri Orlov Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova, Volume 246 (2004), pp. 240-262 | MR 2101296 | Zbl 1101.81093

[32] Jan Šaroch; Jan Šťovíček Singular compactness and definability for Σ-cotorsion and Gorenstein modules, Sel. Math., New Ser., Volume 26 (2020) no. 2, 23, 40 pages | MR 4076700 | Zbl 07186759

[33] Junpeng Wang Ding projective dimension of Gorenstein flat modules, Bull. Korean Math. Soc., Volume 54 (2017) no. 6, pp. 1935-1950 | MR 3733774 | Zbl 1426.16010

[34] Xiaoyan Yang; Nanqing Ding On a question of Gillespie, Forum Math., Volume 27 (2015) no. 6, pp. 3205-3231 | MR 3420339 | Zbl 1347.18003