Model categories from resolution dimensions

Yu, Xuan

doi:10.1090/proc/15069

Model categories from resolution dimensions
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by Xuan Yu PDF

Proc. Amer. Math. Soc. 148 (2020), 3699-3704 Request permission

Abstract:

We discuss model category structures on abelian categories arising from resolution dimensions. In particular, suppose the subcategory of projective objects is an injective cogenerator of $\mathcal {X}$, where $\mathcal {X}$ is a resolving subcategory contained in the subcategory of Gorenstein projective objects. We can conclude that $\mathcal {X}$ is Frobenius and the stable category is given by the homotopy category of a corresponding model structure.

References

Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
Lars Winther Christensen, Hans-Bjørn Foxby, and Henrik Holm, Beyond totally reflexive modules and back: a survey on Gorenstein dimensions, Commutative algebra—Noetherian and non-Noetherian perspectives, Springer, New York, 2011, pp. 101–143. MR 2762509, DOI 10.1007/978-1-4419-6990-3_{5}
Georgios Dalezios, Sergio Estrada, and Henrik Holm, Quillen equivalences for stable categories, J. Algebra 501 (2018), 130–149. MR 3768129, DOI 10.1016/j.jalgebra.2017.12.033
J. R. García Rozas, Covers and envelopes in the category of complexes of modules, Chapman & Hall/CRC Research Notes in Mathematics, vol. 407, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1693036
James Gillespie, Model structures on exact categories, J. Pure Appl. Algebra 215 (2011), no. 12, 2892–2902. MR 2811572, DOI 10.1016/j.jpaa.2011.04.010
Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167–193. MR 2038564, DOI 10.1016/j.jpaa.2003.11.007
Henrik Holm and Peter Jørgensen, Cotorsion pairs in categories of quiver representations, Kyoto J. Math. 59 (2019), no. 3, 575–606. MR 3990178, DOI 10.1215/21562261-2018-0018
Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 1938704, DOI 10.1007/s00209-002-0431-9
Jan Št’ovíček, Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves, Advances in Representation Theory of Algebras, pages 297–367, 2014.