Skip to main content
×
×
Home

Dimensions of triangulated categories

  • Raphaël Rouquier (a1)
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.

Copyright
References
Hide All
Au. Auslander, M., Representation dimension of Artin algebras, Queen Mary College Mathematics Notes, London, 1971
Ba. Backelin, J., On the rate of growth of the homologies of Veronese subrings, Lecture Notes in Math. 1183, 79100, Springer Verlag, 1986
Bei. Beilinson, A.A., Coherent sheaves on P n and problems of linear algebra, Funct. Anal. Appl. 12 (1978), 214216
Bel. Beligiannis, A., Relative homological algebra and purity in triangulated categories, J. Alg. 227 (2000), 268361
BeCaRi. Benson, D.J., Carlson, J.F. and Rickard, J., Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 5980
BöNee. Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209234
BoVdB. Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Moscow Math. J. 3 (2003), 136
ChKeNee. Christensen, J.D., Keller, B. and Neeman, A., Failure of Brown representability in derived categories, Topology 40 (2001), 13391361
CPS. Cline, E., Parshall, B. and Scott, L., Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 8599
CrBo. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 16411674
dNaVdB. de Naeghel, K. and Van den Bergh, M., Ideal classes of three dimensional Sklyanin algebras, J. Algebra 276 (2004), 515551
GeMa. Gelfand, S.I. and Manin, Yu.I., Homological Algebra, Springer Verlag, 1999
Kap. Kapranov, M., On the derived category of coherent sheaves on some homogeneous spaces, Inv. Math. 92 (1988), 479508
Kaw. Kawamata, Y., Equivalences of derived categories of sheaves on smooth stacks, Amer. J. Math. 126 (2004), 10571083
Kr. Krause, H., Auslander-Reiten triangles and a theorem of Zimmermann, Bull. London Math. Soc. 37 (2005), 361372
Ke. Keller, B., Deriving DG Categories, Ann. Sci. Ec. Norm. Sup. 27 (1994), 63102
Ma. Matsumura, H., Commutative Algebra, second edition, Benjamin Cummings, 1980
Nee1. Neeman, A., The chromatic tower for D(R), Topology 31 (1992), 519532
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), 546566
Nee3. Neeman, A., The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205236
Nee4. Neeman, A., On a Theorem of Brown and Adams, Topology 36 (1997), 619645
Nee5. Neeman, A., “Triangulated categories”, Princeton University Press, 2001
Or. Orlov, D., Triangulated categories of singularities and D-branes in Laudau-Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227248
Rou. Rouquier, R., Representation dimension of exterior algebras, Inventiones Math. 165 (2006), 357367
SGA6. Berthelot, P., Grothendieck, A. and Illusie, L., “Théorie des intersections et théorème de Riemann-Roch”, Lecture Notes in Mathematics 225, Springer Verlag, 1971
Ste. Stenström, B., “Rings of quotients”, Springer Verlag, 1975
Th. Thomason, R.W., The classification of triangulated subcategories, Compositio Math. 105 (1997), 127
ThTr. Thomason, R.W. and Trobaugh, T.F., Higher algebraic K-theory of schemes and of derived categories, in “The Grothendieck Festschrift”, vol. III, 247435, Birkhauser, 1990
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 389 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd May 2018. This data will be updated every 24 hours.