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STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS

Published online by Cambridge University Press:  25 February 2013

DAVID BARNES  and
CONSTANZE ROITZHEIM
DAVID BARNES
Affiliation:
The University of Sheffield, School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, United Kingdom e-mail: [email protected]
CONSTANZE ROITZHEIM
Affiliation:
University of Kent, School of Mathematics, Statistics and Actuarial Science, Cornwallis, Canterbury, Kent CT2 7NF, United Kingdom e-mail: [email protected]
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Abstract

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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.

Keywords

55P4255P6018E3016D90
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 13 - 42
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Barnes, D. and Roitzheim, C., Local framings, New York J. Math. 17 (2011), 513–552.Google Scholar
2.Bousfield, A. K., The localization of spaces with respect to homology, Topology 14 (1975), 133–150.Google Scholar
3.Bousfield, A. K., The localization of spectra with respect to homology, Topology 18 (4) (1979), 257–281.Google Scholar
4.Dugger, D., Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353(12) (2001), 5003–5027.CrossRefGoogle Scholar
5.Dugger, D., Spectral enrichments of model categories, Homology Homotopy Appl. 8 (1) (2006), 1–30.CrossRefGoogle Scholar
6.Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (1) (2002), 199–220.CrossRefGoogle Scholar
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.Farjoun, E. D., Cellular spaces, null spaces and homotopy localization, in Lecture Notes in Mathematics, vol. 1622 (Springer, Berlin, 1996).Google Scholar
9.Goerss, P. and Jardine, J., Simplicial homotopy theory, in Progress in Mathematics, vol. 174 (Birkhäuser, Basel, 1999).Google Scholar
10.Hirschhorn, P., Model categories and their localizations, in Mathematical Surveys and Monographs, vol. 99 (American Mathematical Society, Providence, RI, 2003).Google Scholar
11.Hovey, M., Model categories, in Mathematical Surveys and Monographs, vol. 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
12.Hovey, M., Palmieri, J. and Strickland, N., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (610) (1997), x + 114.Google Scholar
13.Lenhardt, F., Stable frames in model categories, J. Pure Appl. Algebra 216 (5) (2012), 1080–1091.Google Scholar
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.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.Schwede, S. and Shipley, B., Stable model categories are categories of modules, Topology 42 (1) (2003), 103–153.Google Scholar