Selecta Mathematica

, Volume 22, Issue 4, pp 2535–2568 | Cite as

Geometricity for derived categories of algebraic stacks

  • Daniel BerghEmail author
  • Valery A. Lunts
  • Olaf M. Schnürer


We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.


Differential graded category Derived category Algebraic stack Root construction Semi-orthogonal decomposition 

Mathematics Subject Classification

Primary 14F05 Secondary 14A20 16E45 



We thank David Rydh for detailed comments. Daniel Bergh was partially supported by Max Planck Institute for Mathematics, Bonn, and by the DFG through SFB/TR 45. Valery Lunts was partially supported by the NSA grant 141008. Olaf Schnürer was partially supported by the DFG through a postdoctoral fellowship and through SPP 1388 and SFB/TR 45.


  1. 1.
    Abramovich, D., Graber, T., Vistoli, A.: Gromov–Witten theory of Deligne–Mumford stacks. Am. J. Math. 130(5), 1337–1398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58(4), 1057–1091 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayer, A., Cadman, C.: Quantum cohomology of \([{\mathbb{C}}^N/\mu _r]\). Compos. Math. 146(5), 1291–1322 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bergh, D.: Functorial destackification of tame stacks with abelian stabilisers. arXiv:1409.5713v1 (2014)
  5. 5.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions/mutations. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183–1205, 1337 (1989)Google Scholar
  6. 6.
    Bergh, D., Rydh, D.: Functorial destackification and weak factorization of orbifolds. In preparation (2015)Google Scholar
  7. 7.
    Bergh, D., Schnürer, O.M.: Conservative descent for semi-orthogonal decompositions. In preparation (2016)Google Scholar
  8. 8.
    Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36, 258 (2003)Google Scholar
  9. 9.
    Cadman, C.: Using stacks to impose tangency conditions on curves. Am. J. Math. 129(2), 405–427 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Choudhury, U.: Motives of Deligne–Mumford stacks. Adv. Math. 231(6), 3094–3117 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Canonaco, A., Stellari, P.: Uniqueness of dg enhancements for the derived category of a Grothendieck category. arxiv:1507.05509v2 (2015)
  12. 12.
    Cisinski, D.-C., Tabuada, G.: Symmetric monoidal structure on non-commutative motives. J. K-Theory 9(2), 201–268 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math 8, 222 (1961)Google Scholar
  14. 14.
    Faltings, G.: Finiteness of coherent cohomology for proper fppf stacks. J. Algebr. Geom. 12(2), 357–366 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fantechi, B., Mann, E., Nironi, F.: Smooth toric Deligne–Mumford stacks. J. Reine Angew. Math. 648, 201–244 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hall, J.: The Balmer spectrum of a tame stack. Ann. K-Theory 1(3), 259–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Herschend, M., Iyama, O.: \(n\)-representation-finite algebras and twisted fractionally Calabi–Yau algebras. Bull. Lond. Math. Soc. 43(3), 449–466 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Halpern-Leistner, D., Pomerleano, D.: Equivariant hodge theory and noncommutative geometry. arXiv:1507.01924v1 (2015)
  19. 19.
    Hall, J., Neeman, A., Rydh, D.: One positive and two negative results for derived categories of algebraic stacks. arXiv:1405.1888v2 (2014)
  20. 20.
    Hall, J., Rydh, D.: Perfect complexes on algebraic stacks. arXiv:1405.1887v2 (2014)
  21. 21.
    Hall, J., Rydh, D.: Algebraic groups and compact generation of their derived categories of representations. Indiana Univ. Math. J. 64, 1903–1923 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    Ishii, A., Ueda, K.: The special McKay correspondence and exceptional collection. arXiv:1104.2381v2 (2011)
  24. 24.
    Keller, B.: On differential graded categories. In: International Congress of Mathematicians. Vol. II, pp. 151–190. Eur. Math. Soc., Zürich (2006)Google Scholar
  25. 25.
    Keel, S., Mori, S.: Quotients by groupoids. Ann. Math. (2) 145(1), 193–213 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kollár, J.: Quotient spaces modulo algebraic groups. Ann. Math. (2) 145(1), 33–79 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kresch, A.: On the geometry of Deligne–Mumford stacks. In: Algebraic geometry—Seattle 2005. Part 1, Volume 80 of Proceedings of Symposium on Pure Mathematics, pp. 259–271. Amer. Math. Soc., Providence, RI (2009)Google Scholar
  28. 28.
    Kresch, A., Vistoli, A.: On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map. Bull. Lond. Math. Soc. 36(2), 188–192 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Laumon, G., Moret-Bailly, L.: Champs algébriques, Volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Berlin (2000)Google Scholar
  30. 30.
    Laszlo, Y., Olsson, M.: The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math Inst. Hautes Études Sci 107, 109–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lunts, V.A., Orlov, D.O.: Uniqueness of enhancement for triangulated categories. J. Am. Math. Soc. 23(3), 853–908 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lunts, V.A., Schnürer, O.M.: Matrix-factorizations and semi-orthogonal decompositions for blowing-ups. J. Noncommut. Geom. arXiv:1212.2670v2 (2012)
  33. 33.
    Lunts, V.A., Schnürer, O.M.: Matrix factorizations and motivic measures. J. Noncommut. Geom. arXiv:1310.7640v2 (2013)
  34. 34.
    Lunts, V.A., Schnürer, O.M.: New enhancements of derived categories of coherent sheaves and applications. J. Algebra 446, 203–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 3rd edn. Springer, Berlin (1994)Google Scholar
  36. 36.
    Olsson, M.: Sheaves on Artin stacks. J. Reine Angew. Math. 603, 55–112 (2007)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Olsson, M.: Integral models for moduli spaces of \(G\)-torsors. Ann. Inst. Fourier (Grenoble) 62(4), 1483–1549 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Orlov, D.O.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 852–862 (1992)zbMATHGoogle Scholar
  39. 39.
    Orlov, D.: Smooth and proper noncommutative schemes and gluing of dg categories. arXiv:1402.7364v5 (2014)
  40. 40.
    Riche, S.: Koszul duality and modular representations of semisimple Lie algebras. Duke Math. J. 154(1), 31–134 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rydh, D.: Existence and properties of geometric quotients. J. Algebr. Geom. 22(4), 629–669 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rydh, D.: Approximation of sheaves on algebraic stacks. Int. Math. Res. Notices 2016(3), 717–737 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rydh, D.: Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia? MathOverflow. (version: 2015-05-09) (2015)
  44. 44.
    Schnürer, O.M.: Six operations on dg enhancements of derived categories of sheaves. arXiv:1507.08697v1 (2015)
  45. 45.
    Alexander, G.: Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Société Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960–61. Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin]Google Scholar
  46. 46.
    Berthelot, P., Grothendieck, A., Illusie, L.: Théorie des intersections et théorème de Riemann-Roch. Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)Google Scholar
  47. 47.
    The Stacks Project Authors. Stacks project. (2016)
  48. 48.
    Tabuada, G.: A guided tour through the garden of noncommutative motives. In: Topics in Noncommutative Geometry, Volume 16 of Clay Mathematics Proceedings, pp. 259–276. Amer. Math. Soc., Providence, RI (2012)Google Scholar
  49. 49.
    Toën, B.: Finitude homotopique des dg-algèbres propres et lisses. Proc. Lond. Math. Soc (3) 98(1), 217–240 (2009)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Toën, B.: Lectures on DG-categories. In: Topics in Algebraic and Topological \(K\)-Theory, Volume 2008 of Lecture Notes in Mathematics, pp. 243–302. Springer, Berlin (2011)Google Scholar
  51. 51.
    Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann Sci École Norm Sup 40(3), 387–444 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Daniel Bergh
    • 1
    Email author
  • Valery A. Lunts
    • 2
  • Olaf M. Schnürer
    • 1
  1. 1.Mathematisches InstitutUniversität BonnBonnGermany
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations