Algebras and Representation Theory

, Volume 17, Issue 4, pp 1181–1206 | Cite as

Perverse Coherent t-Structures Through Torsion Theories

  • Jorge Vitória


Bezrukavnikov, later together with Arinkin, recovered Deligne’s work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes.


t-structure Torsion theory Perverse coherent sheaves Noncommutative projective planes 

Mathematics Subject Classifications (2010)

14F05 13D09 13D30 16E35 16S38 16W50 


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  1. 1.
    Alonso Tarrío, L., Jeremas López, A., Souto Salorio, M.J.: Construction of t-structures and equivalences of derived categories. Trans. Am. Math. Soc. 355(6), 2523–2543 (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alonso Tarrío, L., Jeremas López, A., Saorín, M.: Compactly generated t-structures on the derived category of a Noetherian ring. J. Algebra 324(3), 313–346 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arinkin, D., Bezrukavnikov, R.: Perverse coherent sheaves. Mosc. Math. J. 10(1), 3–29 (2010)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Artin, M., Schelter, W.: Graded algebras of global dimension 3. Adv. Math. 66, 171–216 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift I, pp. 33–85. Birkauser (1990)Google Scholar
  6. 6.
    Artin, M., Tate, J., Van den Bergh, M.: Modules over regular algebras of dimension 3. Invent. Math. 106, 335–388 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Artin, M., Zhang, J.J.: Noncommutative projective schemes. Adv. Math. 109, 228–287 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux Pervers. Asterisque 100 (1982)Google Scholar
  9. 9.
    Bezrukavnikov, R.: Perverse Coherent Sheaves (after Deligne). Arxiv:math.AG/0005152
  10. 10.
    Bondal, A., Van den Bergh, M.: Generators and Representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3, 1–36 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Bridgeland, T.: t-structures on some local Calabi-Yau varities. J. Algebra 289, 453–483 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Colpi, R., Fuller, K.R.: Tilting objects in abelian categories and quasi-tilted rings. Trans. Am. Math. Soc. 359(2), 741–765 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dickson, S.E.: A torsion theory for abelian categories. Trans. Am. Math. Soc. 121, 223–235 (1966)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gabriel, P.: Des catgories abliennes. Bull. Soc. Math. France 90, 323–448 (1962)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Goodearl, K., Stafford, J.T.: The graded version of Goldie’s theorem. Contemp. Math. 259, 237–240 (2000)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Goodearl, K., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings. Cambridge University Press (2004)Google Scholar
  17. 17.
    Happel, D., Reiten, I., Smalø, S.: Tilting in Abelian categories and quasitilted algebras. Memoirs of the American Mathematical Society, vol. 575, viii+88pp (1996)Google Scholar
  18. 18.
    Herstein, I.N.: Noncommutative rings. Carus Math. Monogr., no. 15. Math. Assoc. of America (1968)Google Scholar
  19. 19.
    Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Science Publications (2006)Google Scholar
  20. 20.
    Keller, B., Vossieck, D.: Aisles in derived categories. Bull. Soc. Math. Belg. Sr. A 40(2), 239–253 (1988)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Kashiwara, M.: t-structures on the derived categories of holonomic D-modules and coherent O-modules. Mosc. Math. J. 4(4), 847–868 (2004)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension, vol. 116. Pitman (Advanced Publishing Program) (1985)Google Scholar
  23. 23.
    Lambek, J., Michler, G.: Localization of right noetherian rings at semiprime ideals. Can. J. Math. 26, 1069–1085 (1974)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Matlis, E.: Injective modules over Noetherian rings. Pacific J. Math. 8, 511–528 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Nastasescu, C., Van Oystaeyen, F.: Graded Ring Theory. North-Holland (1982)Google Scholar
  26. 26.
    Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. Research Notes in Mathematics, No. 116. Pitman (1985)Google Scholar
  27. 27.
    Samir Mahmoud, S.: A structure sheaf on the projective spectrum of a graded fully bounded Noetherian ring. Bull. Belg. Math. Soc. Simon Stevin 3, 325–343 (1996)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Serre, J.P.: Faisceaux Algébriques cohérents. Ann. Math. (2) 61, 197–278 (1955)CrossRefzbMATHGoogle Scholar
  29. 29.
    Stanley, D.: Invariants of t-structures and classification of nullity classes. Adv. Math. 224(6), 2662–2689 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Stenström, B.: Rings of Quotients. Springer (1975)Google Scholar
  31. 31.
    Van Oystaeyen, F., Verschoren, A.: Fully bounded Grothendieck categories. II. Graded modules. J. Pure Appl. Algebra 21(2), 189–203 (1981)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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