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Rouquier's Cocovering Theorem and Well-generated Triangulated Categories

Published online by Cambridge University Press:  01 September 2010

Daniel Murfet
Daniel Murfet
Affiliation:
Hausdorff Center for Mathematics, University of Bonn, Germany, [email protected]
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Abstract

We study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal α the condition of α-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is well-generated. As an application we describe the α-compact objects in the unbounded derived category of a quasi-compact and semi-separated scheme.

Keywords

triangulated categorieswell-generated categoriesderived categories
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 1 , August 2011 , pp. 31 - 57
Copyright
Copyright © ISOPP 2010

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References

AJL97.Tarrío, L. Alonso, López, A. Jeremías and Lipman, J., Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 1, 1–39.CrossRefGoogle Scholar
AJS00.Tarrío, L. Alonso, López, A. Jeremías, and Souto Salorio, M. José, Localization in categories of complexes and unbounded resolutions, Canad. J. Math. 52 (2000), no. 2, 225–247.CrossRefGoogle Scholar
AJPV08.Tarrío, L. Alonso, López, A. Jeremías, Rodríguez, M. Pérez, and Gonsalves, M. J. Vale, The derived category of quasi-coherent sheaves and axiomatic stable homotopy, Adv. Math. 218 (2008), no. 4, 1224–1252.CrossRefGoogle Scholar
BN93.Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234.Google Scholar
KS90.Kashiwara, M. and Shapira, P., Sheaves on manifolds, Grundlehren 292, Springer-Verlag, Berlin Heidelberg, 1990.CrossRefGoogle Scholar
Kra01.Krause, H., On Neeman's well generated triangulated categories, Documenta Math. 6 (2001), 121–126.Google Scholar
Kra02.Krause, H., A Brown representability theorem via coherent functors, Topology 41 (2002), 853–861.CrossRefGoogle Scholar
Kra07.Krause, H., Localization theory for triangulated categories, arXiv:0806.1324.Google Scholar
Lip09.Lipman, J., Notes on derived categories and derived functors, Lecture notes in Math. 1960, Springer (2009), 1–259. Available online at: http://www.math.purdue.edu/˜lipman/Duality.pdf.Google Scholar
Mur07.Murfet, D., The mock homotopy category of projectives and Grothendieck duality, Ph.D. thesis, 2007. Available from: http://www.therisingsea.org/thesis.pdf.Google Scholar
Mur08.Murfet, D., The pure derived category of flat sheaves and Grothendieck duality, preprint.Google Scholar
Nee96.Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236.CrossRefGoogle Scholar
Nee01.Neeman, A., Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, Princeton, NJ, 2001.Google Scholar
Nee05.Neeman, A., A survey of well generated triangulated categories, Representations of algebras and related topics, Fields Inst. Commun. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 307–329.Google Scholar
Nee08.Neeman, A., The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2008), 255–308.CrossRefGoogle Scholar
Rou08.Rouquier, R., Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193–256.CrossRefGoogle Scholar
Spa88.Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.Google Scholar
TT90.Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435.CrossRefGoogle Scholar
Ver77.Verdier, J.- L., Catégories derivées. In Séminaire de Géométrie Algébrique du Bois-Marie SGA 4½, Lecture Notes in Math. 569, Springer-Verlag, 1977.Google Scholar