Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T07:21:04.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimensions of triangulated categories

Published online by Cambridge University Press:  30 November 2007

Raphaël Rouquier
Raphaël Rouquier
Affiliation:
[email protected] Institut de Mathématiques de Jussieu — CNRS, UFR de Mathématiques, Université Denis Diderot, 2, place Jussieu, 75005 Paris France
Get access

Abstract

We define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

Keywords

homological algebraderived categoriestriangulated categoriesdimensionsBrown representability
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 2 , April 2008 , pp. 193 - 256
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Au. Auslander, M., Representation dimension of Artin algebras, Queen Mary College Mathematics Notes, London, 1971Google Scholar
Ba. Backelin, J., On the rate of growth of the homologies of Veronese subrings, Lecture Notes in Math. 1183, 79–100, Springer Verlag, 1986Google Scholar
Bei. Beilinson, A.A., Coherent sheaves on Pn and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214–216CrossRefGoogle Scholar
Bel. Beligiannis, A., Relative homological algebra and purity in triangulated categories, J. Alg. 227 (2000), 268–361CrossRefGoogle Scholar
BeCaRi. Benson, D.J., Carlson, J.F. and Rickard, J., Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80CrossRefGoogle Scholar
BöNee. Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234Google Scholar
BoVdB. Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Moscow Math. J. 3 (2003), 1–36CrossRefGoogle Scholar
ChKeNee. Christensen, J.D., Keller, B. and Neeman, A., Failure of Brown representability in derived categories, Topology 40 (2001), 1339–1361CrossRefGoogle Scholar
CPS. Cline, E., Parshall, B. and Scott, L., Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 85–99Google Scholar
CrBo. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1641–1674CrossRefGoogle Scholar
dNaVdB. de Naeghel, K. and Van den Bergh, M., Ideal classes of three dimensional Sklyanin algebras, J. Algebra 276 (2004), 515–551CrossRefGoogle Scholar
GeMa. Gelfand, S.I. and Manin, Yu.I., Homological Algebra, Springer Verlag, 1999Google Scholar
Kap. Kapranov, M., On the derived category of coherent sheaves on some homogeneous spaces, Inv. Math. 92 (1988), 479–508CrossRefGoogle Scholar
Kaw. Kawamata, Y., Equivalences of derived categories of sheaves on smooth stacks, Amer. J. Math. 126 (2004), 1057–1083CrossRefGoogle Scholar
Kr. Krause, H., Auslander-Reiten triangles and a theorem of Zimmermann, Bull. London Math. Soc. 37 (2005), 361–372CrossRefGoogle Scholar
Ke. Keller, B., Deriving DG Categories, Ann. Sci. Ec. Norm. Sup. 27 (1994), 63–102CrossRefGoogle Scholar
Ma. Matsumura, H., Commutative Algebra, second edition, Benjamin Cummings, 1980Google Scholar
Nee1. Neeman, A., The chromatic tower for D(R), Topology 31 (1992), 519–532CrossRefGoogle Scholar
Nee2. 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. Sup. 25 (1992), 546–566Google Scholar
Nee3. Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236CrossRefGoogle Scholar
Nee4. Neeman, A., On a Theorem of Brown and Adams, Topology 36 (1997), 619–645CrossRefGoogle Scholar
Nee5. Neeman, A., “Triangulated categories”, Princeton University Press, 2001CrossRefGoogle Scholar
Or. Orlov, D., Triangulated categories of singularities and D-branes in Laudau-Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227–248Google Scholar
Rou. Rouquier, R., Representation dimension of exterior algebras, Inventiones Math. 165 (2006), 357–367CrossRefGoogle Scholar
SGA6. Berthelot, P., Grothendieck, A. and Illusie, L., “Théorie des intersections et théorème de Riemann-Roch”, Lecture Notes in Mathematics 225, Springer Verlag, 1971Google Scholar
Ste. Stenström, B., “Rings of quotients”, Springer Verlag, 1975CrossRefGoogle Scholar
Th. Thomason, R.W., The classification of triangulated subcategories, Compositio Math. 105 (1997), 1–27CrossRefGoogle Scholar
ThTr. Thomason, R.W. and Trobaugh, T.F., Higher algebraic K-theory of schemes and of derived categories, in “The Grothendieck Festschrift”, vol. III, 247–435, Birkhauser, 1990CrossRefGoogle Scholar