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Mosc. Math. J., 2009, Volume 9, Number 1, Pages 143–149 (Mi mmj340)  

This article is cited in 25 scientific papers (total in 25 papers)

Remarks on generators and dimensions of triangulated categories

Dmitri Orlov

Algebra Section, Steklov Math. Institute of RAS, Moscow, Russia

Abstract: In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.

Key words and phrases: triangulated categories, derived categories, generators, dimension.

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Document Type: Article
MSC: 18E30, 14F05
Received: April 7, 2008
Language: English

Citation: Dmitri Orlov, “Remarks on generators and dimensions of triangulated categories”, Mosc. Math. J., 9:1 (2009), 143–149

Citation in format AMSBIB
\by Dmitri~Orlov
\paper Remarks on generators and dimensions of triangulated categories
\jour Mosc. Math.~J.
\yr 2009
\vol 9
\issue 1
\pages 143--149

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    This publication is cited in the following articles:
    1. M. Abouzaid, I. Smith, “Homological mirror symmetry for the 4-torus”, Duke Math. J., 152:3 (2010), 373–440  crossref  mathscinet  zmath  isi  scopus
    2. S. Oppermann, “The dimension of the derived category of elliptic curves and tubular weighted projective lines”, Colloq. Math., 119:1 (2010), 143–156  crossref  mathscinet  zmath  isi
    3. R. S. Garavuso, L. Katzarkov, M. Kreuzer, A. Noll, “Super Landau-Ginzburg mirrors and algebraic cycles”, J. High Energy Phys., 2011, no. 3, 017, 28 pp.  crossref  mathscinet  zmath  isi  scopus
    4. I. Burban, Yu. Drozd, “Tilting on non-commutative rational projective curves”, Math. Ann., 351:3 (2011), 665–709  crossref  mathscinet  zmath  isi  elib  scopus
    5. M. Ballard, D. Favero, L. Katzarkov, “Orlov spectra: bounds and gaps”, Invent. Math., 189:2 (2012), 359–430  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. M. Ballard, D. Favero, “Hochschild dimensions of tilting objects”, Int. Math. Res. Notices, 2012, no. 11, 2607–2645  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. Alexeev, D. Orlov, “Derived categories of burniat surfaces and exceptional collections”, Math. Ann., 357:2 (2013), 743–759  crossref  mathscinet  zmath  isi  elib  scopus
    8. Ch. Böhning, H.-Ch. G. von Bothmer, P Sosna., “On the derived category of the classical Godeaux surface”, Adv. Math., 243 (2013), 203–231  crossref  mathscinet  zmath  isi  elib  scopus
    9. Katzarkov, G. Kerr, “Orlov spectra as a filtered cohomology theory”, Adv. Math., 243 (2013), 232–261  crossref  mathscinet  zmath  isi  scopus
    10. P. Sosna, “Scalar extensions of triangulated categories”, Appl. Categ. Struct., 22:1 (2014), 211–227  crossref  mathscinet  zmath  isi  elib  scopus
    11. Asadollahi, Javad; Hafezi, Rasool, “On the derived dimension of abelian categories”, Kyoto J. Math., 54:3 (2014), 693–702  crossref  mathscinet  zmath  isi  scopus
    12. Ballard, Matthew; Favero, David; Katzarkov, Ludmil, “A category of kernels for equivariant factorizations, II: further implications”, J. Math. Pures Appl. (9), 102:4 (2014), 702–757  crossref  mathscinet  zmath  isi  scopus
    13. R. Takahashi, “Reconstruction from Koszul homology and applications to module and derived categories”, Pacific J. Math., 268:1 (2014), 231–248  crossref  mathscinet  zmath  isi  elib  scopus
    14. R. Fisette, A. Polishchuk, “$A_\infty$-algebras associated with curves and rational functions on $\mathcal M_{g,g}$. I”, Compos. Math., 150:4 (2014), 621–667  crossref  mathscinet  zmath  isi  elib  scopus
    15. A. Iliev, L. Katzarkov, V. Przyjalkowski, “Double solids, categories and non-rationality”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 145–173  crossref  mathscinet  zmath  isi  scopus
    16. Asadollahi J., Hafezi R., “on the Dimensions of Path Algebras”, Math. Res. Lett., 21:1 (2014), 19–31  crossref  mathscinet  zmath  isi  isi  scopus
    17. Katzarkov L. Liu Y., “Categorical Base Loci and Spectral Gaps, Via Okounkov Bodies and Nevanlinna Theory”, String-Math 2013, Proceedings of Symposia in Pure Mathematics, 88, ed. Donagi R. Douglas M. Kamenova L. Rocek M., Amer Mathematical Soc, 2014, 47–118  crossref  mathscinet  zmath  isi
    18. D. O. Orlov, “Geometric realizations of quiver algebras”, Proc. Steklov Inst. Math., 290:1 (2015), 70–83  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. Aihara T., Takahashi R., “Generators and Dimensions of Derived Categories of Modules”, Commun. Algebr., 43:11 (2015), 5003–5029  crossref  mathscinet  zmath  isi  scopus
    20. Yang S., “a Note on the Rouquier Dimensions of Product Varieties”, J. Algebra. Appl., 15:4 (2016), 1650065  crossref  mathscinet  zmath  isi  scopus
    21. Iyengar S.B. Takahashi R., “Annihilation of Cohomology and Strong Generation of Module Categories”, Int. Math. Res. Notices, 2016, no. 2, 499–535  crossref  mathscinet  zmath  isi  scopus
    22. Burban I., Drozd Yu., Gavran V., “Singular Curves and Quasi-Hereditary Algebras”, Int. Math. Res. Notices, 2017, no. 3, 895–920  crossref  mathscinet  isi  scopus
    23. Lekili Ya. Polishchuk A., “Arithmetic Mirror Symmetry For Genus 1 Curves With N Marked Points”, Sel. Math.-New Ser., 23:3 (2017), 1851–1907  crossref  mathscinet  zmath  isi  scopus
    24. Kikuta K., “On Entropy For Autoequivalences of the Derived Category of Curves”, Adv. Math., 308 (2017), 699–712  crossref  mathscinet  zmath  isi  scopus
    25. D. O. Orlov, “Derived noncommutative schemes, geometric realizations, and finite dimensional algebras”, Russian Math. Surveys, 73:5 (2018), 865–918  mathnet  crossref  crossref  elib
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