Applied Categorical Structures

, Volume 19, Issue 5, pp 757–782 | Cite as

Perfect Derived Categories of Positively Graded DG Algebras

  • Olaf M. SchnürerEmail author


We investigate the perfect derived category \({{\rm dgPer}}(\mathcal{A})\) of a positively graded differential graded (dg) algebra \(\mathcal{A}\) whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of \({{{\rm dgPer}}}(\mathcal{A})\) whose objects are easy to describe, define a t-structure on \({{{\rm dgPer}}}(\mathcal{A})\) and study its heart. We show that \({{{\rm dgPer}}}(\mathcal{A})\) is a Krull–Remak–Schmidt category. Then we consider the heart in the case that \(\mathcal{A}\) is a Koszul ring with differential zero satisfying some finiteness conditions.


Differential graded module DG module t-structure Heart Koszul duality 

Mathematics Subject Classifications (2000)

18E30 16D90 


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  1. 1.
    Avramov, L.L., Foxby, H.-B., Halperin, S.: Resolutions for dg Modules (2008, preliminary version)Google Scholar
  2. 2.
    Avramov, L.L., Martsinkovsky, A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. London Math. Soc. (3), 85(2), 393–440 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auslander, M.: Representation theory of Artin algebras. I. Comm. Algebra 1, 177–268 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auslander, M.: Representation theory of Artin algebras. II. Comm. Algebra 1, 269–310 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris, (1982)Google Scholar
  6. 6.
    Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9(2), 473–527 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernstein, J., Lunts, V.: Equivariant sheaves and functors. Lecture Notes in Mathematics, vol. 1578. Springer, Berlin (1994)Google Scholar
  8. 8.
    Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compositio Math. 86(2), 209–234 (1993)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bott, R., Tu, L.W.: Differential forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)Google Scholar
  11. 11.
    Chen, X.-W., Ye, Y., Zhang, P.: Algebras of derived dimension zero. Comm. Algebra 36(1), 1–10 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guillermou, S.: Equivariant derived category of a complete symmetric variety. Represent. Theory 9, 526–577 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994)zbMATHGoogle Scholar
  16. 16.
    Keller, B.: On the construction of triangle equivalences. In: Derived equivalences for group rings. Lecture Notes in Math., vol. 1685, pp. 155–176. Springer, Berlin (1998)CrossRefGoogle Scholar
  17. 17.
    Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Krause, H.: Krull-Remak-Schmidt categories and projective covers. Note, pp. 1–9. (2008)
  19. 19.
    Le, J., Chen, X. -W.: Karoubianness of a triangulated category. J. Algebra 310(1), 452–457 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lunts, V.: Equivariant sheaves on toric varieties. Compositio Math. 96(1), 63–83 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mazorchuk, V., Ovsienko, S., Stroppel, C.: Quadratic duals, Koszul dual functors, and applications. Trans. Amer. Math. Soc. 361(3), 1129–1172 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ringel, C.M.: Tame algebras and integral quadratic forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)Google Scholar
  23. 23.
    Schnürer, O.M.: Equivariant sheaves on flag varieties. Math. Z. doi: 10.1007/s00209-009-0609-5
  24. 24.
    Schnürer, O.M.: Equivariant Sheaves on Flag Varieties, DG Modules and Formality. Doktorarbeit, Universität Freiburg. (2007)

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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