Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T08:24:58.387Z Has data issue: false hasContentIssue false
  • English
  • Français

THE DERIVED CATEGORY OF AN ÉTALE EXTENSION AND THE SEPARABLE NEEMAN–THOMASON THEOREM

Part of: Abelian categories (Co)homology theory

Published online by Cambridge University Press:  12 December 2014

Paul Balmer
Paul Balmer*
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA ([email protected]) URL: http://www.math.ucla.edu/∼balmer
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that étale morphisms of schemes yield separable extensions of derived categories. We then generalize the Neeman–Thomason localization theorem to separable extensions of triangulated categories.

Keywords

Neeman–Thomasonseparable extensionétalederived category

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 3 , July 2016 , pp. 613 - 623
Copyright
© Cambridge University Press 2014 

References

Balmer, P., Separability and triangulated categories, Adv. Math. 226(5) (2011), 4352–4372.Google Scholar
Balmer, P., Stacks of group representations, J. Eur. Math. Soc. (JEMS) (in press), Preprint, 2013, available at arXiv:1302.6290.Google Scholar
Balmer, P., Dell’Ambrogio, I. and Sanders, B., Restriction to subgroups as étale extensions, in topology, KK-theory and geometry, Preprint, 2014, available online atarXiv:1408.2524.Google Scholar
Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compos. Math. 86(2) (1993), 209–234.Google Scholar
Bondal, A. and van den Bergh, M., Generators and representability of functors in commutative and non-commutative geometry, Mosc. Math. J. 3(1) (2003), 1–36, 258.Google Scholar
Eilenberg, S. and Moore, J. C., Adjoint functors and triples, Illinois J. Math. 9 (1965), 381–398.CrossRefGoogle Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique I (Springer-Verlag, Berlin, 1971).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. (32) (1967).Google Scholar
Hall, J. and Rydh, D., Perfect complexes on algebraic stacks, Preprint, 2014, available at arXiv:1405.1887.Google Scholar
Hovey, M., Palmieri, J. H. and Strickland, N. P., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128(610) (1997), x+114 pp.Google Scholar
Krause, H., A Brown representability theorem via coherent functors, Topology 41(4) (2002), 853–861.Google Scholar
Krause, H., Localization for triangulated categories, in Triangulated Categories, London Mathematical Society Lecture Note Series, Volume 375, pp. 161–235 (Cambridge University Press, Cambridge, 2010).Google Scholar
Künzer, M., Heller triangulated categories, Homology, Homotopy Appl. 9(2) (2007), 233–320.Google Scholar
Lipman, J., Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics, Volume 1960, pp. 1–259 (Springer, Berlin, 2009).Google Scholar
Maltsiniotis, G., Catégories triangulées supérieures, Preprint, 2006, available online at http://people.math.jussieu.fr/∼maltsin/ps/triansup.ps.Google Scholar
Miller, H., Finite localizations, Bol. Soc. Mat. Mexicana (2) 37(1–2) (1992), 383–389. Papers in honor of José Adem (Spanish).Google Scholar
Mac Lane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics, Volume 5 (Springer-Verlag, New York, 1998).Google Scholar
Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. (4) 25(5) (1992), 547–566.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9(1) (1996), 205–236.Google Scholar
Neeman, A., Triangulated Categories, Annals of Mathematics Studies, Volume 148 (Princeton University Press, 2001).Google Scholar
Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III, Progress in Mathematics, Volume 88, pp. 247–435 (Birkhäuser, Boston, MA, 1990).Google Scholar