Geodesic
STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS
Glasgow mathematical journal, Tome 56 (2014) no. 1, pp. 13-42

Voir la notice de l'article provenant de la source Cambridge University Press

We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.
DOI : 10.1017/S0017089512000882
Mots-clés : 55P42, 55P60, 18E30, 16D90 
BARNES, DAVID; ROITZHEIM, CONSTANZE. STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS. Glasgow mathematical journal, Tome 56 (2014) no. 1, pp. 13-42. doi: 10.1017/S0017089512000882
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[1] 1.Barnes, D. and Roitzheim, C., Local framings, New York J. Math. 17 (2011), 513–552. Google Scholar

[2] 2.Bousfield, A. K., The localization of spaces with respect to homology, Topology 14 (1975), 133–150. Google Scholar

[3] 3.Bousfield, A. K., The localization of spectra with respect to homology, Topology 18 (4) (1979), 257–281. Google Scholar

[4] 4.Dugger, D., Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353(12) (2001), 5003–5027. Google Scholar | DOI

[5] 5.Dugger, D., Spectral enrichments of model categories, Homology Homotopy Appl. 8 (1) (2006), 1–30. Google Scholar | DOI

[6] 6.Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (1) (2002), 199–220. Google Scholar | DOI

[7] 7.Elmendorf, A. D., Kriz, I., Mandell, M. A. and May, J. P., Rings, modules, and algebras in stable homotopy theory, in Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997). With an appendix by M. Cole. Google Scholar

[8] 8.Farjoun, E. D., Cellular spaces, null spaces and homotopy localization, in Lecture Notes in Mathematics, vol. 1622 (Springer, Berlin, 1996). Google Scholar

[9] 9.Goerss, P. and Jardine, J., Simplicial homotopy theory, in Progress in Mathematics, vol. 174 (Birkhäuser, Basel, 1999). Google Scholar

[10] 10.Hirschhorn, P., Model categories and their localizations, in Mathematical Surveys and Monographs, vol. 99 (American Mathematical Society, Providence, RI, 2003). Google Scholar

[11] 11.Hovey, M., Model categories, in Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999). Google Scholar

[12] 12.Hovey, M., Palmieri, J. and Strickland, N., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (610) (1997), x + 114. Google Scholar

[13] 13.Lenhardt, F., Stable frames in model categories, J. Pure Appl. Algebra 216 (5) (2012), 1080–1091. Google Scholar

[14] 14.Ravenel, D., Life after the telescope conjecture, in Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407 (Kluwer, Dordrecht, 1993), 205–222. Google Scholar

[15] 15.Schwede, S. and Shipley, B., Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2) (2000), 491–511. Google Scholar

[16] 16.Schwede, S. and Shipley, B., Stable model categories are categories of modules, Topology 42 (1) (2003), 103–153. Google Scholar

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