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Brown representability often fails for homotopy categories of complexes

Published online by Cambridge University Press:  07 November 2011

George Ciprian Modoi  and
Jan Šťovíček
George Ciprian Modoi
Affiliation:
Babeş–Bolyai University, Faculty of Mathematics and Computer Science, 1, Mihail Kogălniceanu, 400084 Cluj–Napoca, Romaniacmodoi@math.ubbcluj.ro
Jan Šťovíček
Affiliation:
Charles University in Prague, Faculty of Mathematics and Physics, Department of Algebra, Sokolovska 83, 186 75 Praha 8, Czech Republicstovicek@karlin.mff.cuni.cz
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Abstract

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.

Keywords

Brown representabilityAdjoint functorTriangulated categoryHomotopy category of complexes
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 1 , February 2012 , pp. 151 - 160
Copyright
Copyright © ISOPP 2011

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References

1.Azumaya, G. and Facchini, A.. Rings of pure global dimension zero and Mittag-Leffler modules. J. Pure Appl. Algebra 62(2):109–122, 1989.CrossRefGoogle Scholar
2.Balmer, P. and Schlichting, M.. Idempotent completion of triangulated categories. J. Algebra 236(2):819–834, 2001.CrossRefGoogle Scholar
3.Bazzoni, S. and Šťovíček, J.. Flat Mittag-Leffler modules over countable rings. To appear in Proc. Amer. Math. Soc., arXiv:1007.4977v2, 2010.Google Scholar
4.Bravo, D., Enochs, E., Iacob, A., Jenda, O., and Rada, J.. Cotorsion pairs in C(R-Mod). To appear in Rocky Mountain J. Math., 2010.Google Scholar
5.Casacuberta, C. and Neeman, A.. Brown representability does not come for free. Math. Res. Lett. 16(1):1–5, 2009.CrossRefGoogle Scholar
6.Chase, S. U.. Direct products of modules. Trans. Amer. Math. Soc. 97:457–473, 1960.CrossRefGoogle Scholar
7.Eklof, P. C. and Mekler, A. H.. Almost free modules, North-Holland Mathematical Library 65. North-Holland Publishing Co., Amsterdam, revised edition, 2002. Settheoretic methods.Google Scholar
8.Estrada, S., Asensio, P. Guil, Prest, M., and Trlifaj, J.. Model category structures arising from Drinfeld vector bundles. Preprint, arXiv:0906.5213v1, 2009.Google Scholar
9.Fuchs, L.. Infinite abelian groups. Pure and Applied Mathematics 36-II. Academic Press, New York, 1973.Google Scholar
10.Herbera, D. and Trlifaj, J.. Almost free modules and Mittag-Leffler conditions. Preprint, arXiv:0910.4277v1, 2009.Google Scholar
11.Krause, H.. Approximations and adjoints in homotopy categories. To appear in Math. Ann., arXiv:1005.0209v2, 2010.Google Scholar
12.Krause, H.. Localization theory for triangulated categories. In Triangulated categories, London Math. Soc. Lecture Note Ser. 375, pages 161–235. Cambridge Univ. Press, Cambridge, 2010.CrossRefGoogle Scholar
13.Murfet, D.. The Mock Homotopy Category of Projectives and Grothendieck Duality. PhD thesis, Australian National University, 2007. Available at http://www.therisingsea.org/thesis.pdf.Google Scholar
14.Neeman., A.Triangulated categories, Annals of Mathematics Studies 148. Princeton University Press, Princeton, NJ, 2001.CrossRefGoogle Scholar
15.Neeman, A.. The homotopy category of flat modules, and Grothendieck duality. Invent. Math. 174(2):255–308, 2008.CrossRefGoogle Scholar
16.Neeman, A.. Some adjoints in homotopy categories. Ann. of Math. (2), 171(3):2143–2155, 2010.CrossRefGoogle Scholar
17.Raynaud, M. and Gruson, L.. Critères de platitude et de projectivité. Techniques de “platification” d'un module. Invent. Math. 13:1–89, 1971.CrossRefGoogle Scholar
18.Saorín, M. and Šťovíček, J.. On exact categories and applications to triangulated adjoints and model structures. Adv. Math. 228(2):968–1007, 2011.CrossRefGoogle Scholar
19.Šaroch, J. and Trlifaj, J.. Kaplansky classes, finite character, and א1-projectivity. To appear in Forum Math., doi:10.1515/FORM.2011.101.CrossRefGoogle Scholar
20.Šťovíček, J.. Locally well generated homotopy categories of complexes. Doc. Math. 15:507–525, 2010.CrossRefGoogle Scholar
21.Walker, E. A.. The groups P β. In Symposia Mathematica XIII, (Convegno di Gruppi Abeliani, INDAM, Rome, 1974), pages 245–255. Academic Press, London, 1974.Google Scholar