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Stable left and right Bousfield localisations. (English) Zbl 1297.55020
Many interesting model categories arise by enlarging the class of weak equivalences in a given (simpler) model category $$\mathcal C$$. Often this is achieved by forming the left Bousfield localization of $$\mathcal C$$ with respect to a class of maps $$S$$. Left Bousfield localizations are known to exist if $$\mathcal C$$ and $$S$$ satisfy suitable assumptions, see e.g. [P. S. Hirschhorn, Model categories and their localizations. Mathematical Surveys and Monographs. 99. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)].
In the paper under review, the authors define a class of maps $$S$$ in a stable model category $$\mathcal C$$ to be stable if the $$S$$-local objects are closed under suspension. The main results state that if in this situation the left Bousfield localization exists, then it has various desirable properties: It is again a stable model structure, it is right proper if $$\mathcal C$$ is, it has an explicit set of generating acyclic cofibrations if $$\mathcal C$$ is proper and cellular, and it interacts well with monoidal products. The authors also establish corresponding results for right Bousfield localizations and discuss applications to Morita theory and spectral model categories.

##### MSC:
 55U35 Abstract and axiomatic homotopy theory in algebraic topology 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory 18E30 Derived categories, triangulated categories (MSC2010) 16D90 Module categories in associative algebras
##### Keywords:
Bousfield localization; stable model category
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##### References:
 [1] Hovey, Mem. Amer. Math. Soc. 128 pp 114– (1997) [2] DOI: 10.1016/j.jpaa.2011.10.024 · Zbl 1273.55004 · doi:10.1016/j.jpaa.2011.10.024 [3] Hirschhorn, Mathematical Surveys and Monographs (2003) [4] Goerss, Progress in Mathematics (1999) [5] Farjoun, Lecture Notes in Mathematics (1996) [6] Elmendorf, Mathematical Surveys and Monographs (1997) [7] DOI: 10.1353/ajm.2002.0001 · Zbl 1017.18008 · doi:10.1353/ajm.2002.0001 [8] Dugger, Homology Homotopy Appl. 8 pp 1– (2006) · Zbl 1084.55011 · doi:10.4310/HHA.2006.v8.n1.a1 [9] DOI: 10.1090/S0002-9947-01-02661-7 · Zbl 0974.55011 · doi:10.1090/S0002-9947-01-02661-7 [10] DOI: 10.1016/S0040-9383(02)00006-X · Zbl 1013.55005 · doi:10.1016/S0040-9383(02)00006-X [11] DOI: 10.1016/0040-9383(79)90018-1 · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1 [12] DOI: 10.1112/S002461150001220X · Zbl 1026.18004 · doi:10.1112/S002461150001220X [13] DOI: 10.1016/0040-9383(75)90023-3 · Zbl 0309.55013 · doi:10.1016/0040-9383(75)90023-3 [14] Ravenel, Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. pp 205– (1993) · doi:10.1007/978-94-017-0695-7_10 [15] Barnes, New York J. Math. 17 pp 513– (2011) [16] Hovey, Mathematical Surveys and Monographs (1999)
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