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t-structures on hereditary categories

  • Donald Stanley
  • Adam-Christiaan van Roosmalen
Article
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Abstract

We study aisles, equivalently t-structures, in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t-structures in the derived category. In some cases, including the case of finite-dimensional modules over a finite-dimensional hereditary algebra, we refine our results and show that the t-structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring.

Keywords

Hereditary categories Derived categories T-structures Wide subcategories 

Mathematics Subject Classification

18E30 18A40 

Notes

Acknowledgements

The authors would like to thank Jan Šťovíček for meaningful discussion and especially for pointing out Proposition 2.13 to us. The first author wishes to thank the Max-Planck-Institut in Bonn for their scientific support. The second author also gratefully acknowledges the support of the Hausdorff Center for Mathematics in Bonn and Bielefeld University, during which part of this article has been written.

References

  1. 1.
    Alonso Tarrío, L., López, A.J., Saorín, M.: Compactly generated \(t\)-structures on the derived category of a Noetherian ring. J. Algebra 324(3), 313–346 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arinkin, D., Bezrukavnikov, R.: Perverse coherent sheaves. Mosc. Math. J. 10(1), 3–29 (2010). 271MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceauxpervers, Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, pp. 5–171 (1982)Google Scholar
  4. 4.
    Bondal, A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989)MathSciNetGoogle Scholar
  5. 5.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183–1205 (1989). 1337Google Scholar
  6. 6.
    Borceux, F.: Handbook of Categorical Algebra. 1, Encyclopedia of Mathematics and Its Applications. Basic Category Theory, vol. 50. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  7. 7.
    Brüning, K.: Thick subcategories of the derived category of a hereditary algebra. Homol. Homotopy Appl. 9(2), 165–176 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burban, I., Kreussler, B.: Derived categories of irreducible projective curves of arithmetic genus one. Compos. Math. 142, 1231–1262 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, X.-W.: Extensions of covariantly finite subcategories. Arch. Math. (Basel) 93(1), 29–35 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, X.-W., Krause, H.: Introduction to coherent sheaves on weighted projective lines (preprint)Google Scholar
  11. 11.
    Dichev, N.D.: Thick Subcategories for Quiver Representations. Ph.D. Thesis, University of Paderborn (2009)Google Scholar
  12. 12.
    Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algeabra 144(2), 273–343 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra, Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2013)Google Scholar
  14. 14.
    Gorodentsev, A.L., Kuleshov, S.A., Rudakov, A.N.: \(t\)-stabilities and \(t\)-structures on triangulated categories. Izv. Ross. Akad. Nauk Ser. Mat. 68(4), 117–150 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575), viii+ 88 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hartshorne, R.: Residues and Dualities. Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64. With An Appendix by P. Deligne. Springer, Berlin (1966)Google Scholar
  17. 17.
    Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer, New York (1977)Google Scholar
  18. 18.
    Ingalls, C., Thomas, H.: Noncrossing patitions and representations of quivers. Compos. Math. 145, 1533–1562 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jacobson, N.: Basic Algebra. II, 2nd edn. W. H. Freeman and Company, New York (1989)zbMATHGoogle Scholar
  20. 20.
    Keller, B., Vossieck, D.: Aisles in derived categories. Bull. Soc. Math. Belg. Sér. A 40(2), 239–253 (1988). Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Keller, B.: Derived Categories and Tilting, Handbook of Tilting Theory (Hügel, Lidia Angeleri, Happel, Dieter Krause, Henning eds.), London Mathematical Society Lecture Notes Series, vol. 332, pp. 49–104. Cambridge University Press, Cambridge (2007)Google Scholar
  22. 22.
    Krause, H.: Report on Locally Finite Triangulated Categories. arXiv:1107.2631
  23. 23.
    Liu, Q., Vitória, J.: \(t\)-structures via recollements for piecewise hereditary algebras. J. Pure Appl. Algebra 216(4), 837–849 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    MacLane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer, New York (1971)Google Scholar
  25. 25.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings, revised ed., Graduate Studies in Mathematics, vol. 30. With the cooperation of L. W. Small. American Mathematical Society, Providence (2001)Google Scholar
  26. 26.
    Neeman, A.: Some adjoints in homotopy categories. Ann. Math. (2) 171(3), 2143–2155 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Parthasarathy, R.: \(t\)-structures in the derived category of representations of quivers. Proc. Indian Acad. Sci. Math. Sci. 98(2–3), 187–214 (1988)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Polishchuk, A.: Noncommutative proj and coherent algebras. Math. Res. Lett. 12(1), 63–74 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Salorio, S., José, M., Trepode, S.: T-structures on the bounded derived category of the Kronecker algebra. Appl. Categ. Struct. 20, 513–529 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Stanley, D.: Invariants of \(t\)-structures and classification of nullity classes. Adv. Math. 224(6), 2662–2689 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stanley, D., Wang, B.: Classifying subcategories of finitely generated modules over a noetherian ring. J. Pure Appl. Algebra 215(11), 2684–2693 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Van den Bergh, M.: Abstract blowing down. Proc. Am. Math. Soc. 128(2), 375–381 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    van Roosmalen, A.-C.: Abelian 1-Calabi-Yau categories. Int. Math. Res. Not. 2008, 9 (2008)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Donald Stanley
    • 1
  • Adam-Christiaan van Roosmalen
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada
  2. 2.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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