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Cellularization of structures in stable homotopy categories

Published online by Cambridge University Press:  17 May 2012

JAVIER J. GUTIÉRREZ
JAVIER J. GUTIÉRREZ*
Affiliation:
Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain. e-mail: [email protected]
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Abstract

We describe the formal framework for cellularization functors in triangulated categories and study the preservation of ring and module structures under these functors in stable homotopy categories in the sense of Hovey, Palmieri and Strickland, such as the homotopy category of spectra or the derived category of a commutative ring. We prove that cellularization functors preserve modules over connective rings but they do not preserve rings in general (even if the ring is connective or the cellularization functor is triangulated). As an application of these results, we describe the cellularizations of Eilenberg–Mac Lane spectra and compute all acyclizations in the sense of Bousfield of the integral Eilenberg–Mac Lane spectrum.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 399 - 418
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Adams, J. F.Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics, University of Chicago Press, 1974.Google Scholar
[2]Alonso, L.Tarrío, A.López, Jeremías and Salorio, M. J. SoutoConstruction of t-structures and equivalences of derived categories. Trans. Amer. Math. Soc. 355 (2003), 2523–2543.CrossRefGoogle Scholar
[3]Barwick, C.On left and right model categories and left and right Bousfield localizations. Homology, Homotopy Appl. 12 (2) (2010), 245–320.CrossRefGoogle Scholar
[4]Beĭlinson, A. A., Bernstein, J. and Deligne, P.Faisceaux pervers. Astérisque 100, 1982.Google Scholar
[5]Bousfield, A. K.The localization of spectra with respect to homology. Topology 18 (1979), no 4, 257–281.CrossRefGoogle Scholar
[6]Bousfield, A. K.The Boolean algebra of spectra. Comment. Math. Helv. 54 (1979), no. 3, 368–377.CrossRefGoogle Scholar
[7]Casacuberta, C. and Gutiérrez, J. J.Homotopical localization of module spectra. Trans. Amer. Math. Soc. 357 (2005), no. 7, 2753–2770.CrossRefGoogle Scholar
[8]Casacuberta, C., Gutiérrez, J. J. and Rosický, J. Are all localizing subcategories of stable homotopy categories coreflective? Preprint arXiv:1106.2218.Google Scholar
[9]Chachólski, W.On the functors CellA and P A. Duke. Math. J. 84 (1996), no. 3, 599–631.CrossRefGoogle Scholar
[10]Chachólski, W.Desuspending and delooping cellular inequalities. Invent. Math. 129 (1997), no. 1, 37–62.Google Scholar
[11]Chachólski, W., Farjoun, E. D., Göbel, R. and Segev, Y.Cellular covers of divisible abelian groups. In: Alpine Perspectives on Algebraic Topology, Contemp. Math. vol. 504 (Amer. Math. Soc., Providence 2009), 77–97.CrossRefGoogle Scholar
[12]Chachólski, W., Parent, P.-E. and Stanley, D.Cellular generators. Proc. Amer. Math. Soc. 132 (2004), no. 11, 3397–3409.CrossRefGoogle Scholar
[13]Chorny, B.Abstract cellularization as a cellularization with respect to a set of objects. In: Categories in Algebra, Geometry and Mathematical Physics (Sydney 2005), Contemp. Math. vol. 431 (Amer. Math. Soc., Providence 2007), 165–170.CrossRefGoogle Scholar
[14]Farjoun, E. D.Cellular Spaces, Null Spaces and Homotopy Localization. Lecture Notes in Math. vol 1622 (Springer-Verlag, 1996).CrossRefGoogle Scholar
[15]Farjoun, E. D., Göbel, R. and Segev, Y.Cellular covers of groups. J. Pure Appl. Algebra 208 (2007), 61–76.CrossRefGoogle Scholar
[16]Fuchs, L. and Göbel, R.Cellular covers of abelian groups. Results Math. 53 (2009), 59–76.CrossRefGoogle Scholar
[17]Göbel, R., Rodríguez, J. L. and Strüngmann, L. Cellular covers of cotorsion-free modules. To appear in Fund. Math.Google Scholar
[18]Gutiérrez, J. J.Strict modules and homotopy modules in stable homotopy. Homology Homotopy Appl. 7 (2005), no. 1, 39–49.CrossRefGoogle Scholar
[19]Gutiérrez, J. J.Homological localizations of Eilenberg–MacLane spectra. Forum Math. 22 (2010), no. 2, 349–356.CrossRefGoogle Scholar
[20]Hirschhorn, P. S.Model Categories and their Localizations. Mathematical Surveys and Monographs vol. 99 (American Mathematical Society, 2003).Google Scholar
[21]Hovey, M.Model Categories. Mathematical Surveys and Monographs vol. 63 (American Mathematical Society, 1999).Google Scholar
[22]Hovey, M., Palmieri, J. H. and Strickland, N. P.Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128 (1997).Google Scholar
[23]Hovey, M., Shipley, B. and Smith, J.Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.CrossRefGoogle Scholar
[24]Kaplansky, I.Infinite Abelian Groups (The University of Michigan Press, Ann Arbor, 1969).Google Scholar
[25]Neeman, A.Triangulated Categories. Annals of Mathematical Studies vol. 148 (Princeton University Press, 2001).CrossRefGoogle Scholar
[26]Verdier, J. L.Catégories Dérivées. Cohomologie Etale (SGA 4), Lecture Notes in Math. vol. 596 (Springer-Verlag, 1977), pp 262–311.Google Scholar