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Algebras and Representation Theory

, Volume 21, Issue 4, pp 737–767 | Cite as

Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

  • Leonid Positselski
Article
  • 23 Downloads

Abstract

We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras \(\mathcal {C}\) and \(\mathcal {D}\). Given such a complex \(\mathcal {B}^{\bullet }\), we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over \(\mathcal {C}\) and left contramodules over \(\mathcal {D}\). Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material.

Keywords

Artinian coalgebras Co-Noetherian coalgebras Cocoherent coalgebras Comodules Contramodules Semialgebras Semimodules Semicontramodules Derived categories Derived comodule-contramodule correspondence MGM duality/equivalence Dedualizing complexes 

Mathematics Subject Classification (2010)

16T15 16D90 16E35 16P70 

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References

  1. 1.
    Ballard, M., Deliu, D., Favero, D., Isik, M.U., Katzarkov, L.: Resolutions in factorization categories. Adv. Math. 295, 195–249 (2016). arXiv:1212.3264 [math.CT]MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bernstein, J., Lunts, V.: Equivariant sheaves and functors. Lecture Notes in Math, vol. 1578. Springer-Verlag, Berlin (1994)MATHGoogle Scholar
  3. 3.
    Christensen, L.W., Frankild, A., Holm, H.: On Gorenstein projective, injective, and flat dimensions—A functorial description with applications. J. Algebra 302(1), 231–279 (2006). arXiv:0403156 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dwyer, W.G., Greenlees, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Efimov, A.I., Positselski, L.: Coherent analogues of matrix factorizations and relative singularity categories. Algebra and Number Theory 9(5), 1159–1292 (2015). arXiv:1102.0261 [math.CT]MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gómez-Torrecillas, J., Năstăsescu, C., Torrecillas, B.: Localization in coalgebras. Applications to finiteness conditions. J. Algebra Appl. 6(2), 233–243 (2007). arXiv:0403248 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Greenlees, J.P.C., May, J.P.: Derived functors of I-adic completion and local homology. J. Algebra 149(2), 438–453 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Harrison, D.K.: Infinite abelian groups and homological methods. Ann. Math. 69(2), 366–391 (1959)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hinich, V.: Homological algebra of homotopy algebras. Comm. Algebra 25(10), 3291–3323 (1997). arXiv:9702015. Erratum, arXiv:0309453 [math.QA]MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hinich, V.: DG coalgebras as formal stacks. J Pure Appl. Algebra 162(2–3), 209–250 (2001). arXiv:9812034 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Husemoller, D., Moore, J.C., Stasheff, J.: Differential homological algebra and homogeneous spaces. J. Pure Appl. Algebra 5(2), 113–185 (1974)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Iyengar, S., Krause, H.: Acyclicity versus total acyclicity for complexes over noetherian rings. Documenta Math 11, 207–240 (2006)MathSciNetMATHGoogle Scholar
  13. 13.
    Jørgensen, P.: The homotopy category of complexes of projective modules. Adv. Math. 193(1), 223–232 (2005). arXiv:0312088 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kaledin, D.: Derived Mackey functors. Moscow Math. J. 11(4), 723–803 (2011). arXiv:0812.2519 [math.KT]MathSciNetMATHGoogle Scholar
  15. 15.
    Keller, B.: Deriving DG-categories. Ann. Sci. de l’École Norm. Sup. (4) 27(1), 63–102 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Keller, B.: Koszul duality and coderived categories (after K. Lefèvre). October 2003. Available from. http://www.math.jussieu.fr/∼keller/publ/index.html
  17. 17.
    Krause, H.: The stable derived category of a Noetherian scheme. Compositio Math. 141(5), 1128–1162 (2005). arXiv:0403526 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lefèvre-Hasegawa, K.: Sur les A-catégories. Thèse de doctorat, Université Denis Diderot – Paris 7. arXiv:0310337. Corrections, by B. Keller. Available from http://people.math.jussieu.fr/∼keller/lefevre/publ.html (2003)
  19. 19.
    Matlis, E.: Cotorsion modules. Memoirs of the American Math. Society 49 (1964)Google Scholar
  20. 20.
    Matlis, E.: The higher properties of R-sequences. J. Algebra 50(1), 77–112 (1978)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Miyachi, J.: Derived categories and Morita duality theory. J. Pure Appl. Algebra 128(2), 153–170 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Orlov, D.: Matrix factorizations for nonaffine LG-models. Mathematische Annalen 353(1), 95–108 (2012). arXiv:1101.4051 [math.AG]MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Porta, M., Shaul, L., Yekutieli, A.: On the homology of completion and torsion. Algebras and Representation Theory 17(1), 31–67 (2014). arXiv:1010.4386 [math.AC]. Erratum in Algebras and Representation Theory 18, #5, p. 1401–1405, 2015. arXiv:1506.07765 [math.AC]MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Positselski, L.: Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne vol. 70, Birkhäuser/Springer Basel. xxiv+349 pp. arXiv:0708.3398 [math.CT] (2010)
  25. 25.
    Positselski, L.: Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Memoirs Am. Math. Soc. 212, 996 (2011). vi+133 pp. arXiv:0905.2621 [math.CT]MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Positselski, L.: Weakly curved A -algebras over a topological local ring. Electronic preprint. arXiv:1202.2697 [math.CT]
  27. 27.
    Positselski, L.: Contraherent cosheaves. Electronic preprint. arXiv:1209.2995 [math.CT]
  28. 28.
    Positselski, L.: Contramodules. Electronic preprint. arXiv:1503.00991 [math.CT]
  29. 29.
    Positselski, L.: Dedualizing complexes and MGM duality. J. Pure Appl. Algebra 220(12), 3866–3909 (2016). arXiv:1503.05523 [math.CT]MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Positselski, L.: Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre–Grothendieck duality. Selecta Math. (New Ser.) 23(2), 1279–1307 (2017). arXiv:1504.00700 [math.CT]MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Positselski, L.: Contraadjusted modules, contramodules, and reduced cotorsion modules. Moscow Math. J. 17(3), 385–455 (2017). arXiv:1605.03934 [math.CT]MathSciNetGoogle Scholar
  32. 32.
    Positselski, L.: Triangulated Matlis equivalence. Electronic preprint. arXiv:1605.08018 [math.CT], to appear in Journ. of Algebra and its Appl.
  33. 33.
    Positselski, L.: Smooth duality and co-contra correspondence. Electronic preprint arXiv:1609.04597 [math.CT]
  34. 34.
    Spaltenstein, N.: Resolutions of unbounded complexes. Compositio Math. 65(2), 121–154 (1988)MathSciNetMATHGoogle Scholar
  35. 35.
    Sweedler, M.E.: Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc., New York (1969)Google Scholar
  36. 36.
    Takeuchi, M.: Morita theorems for categories of comodules. J Faculty Scie. Univ. Tokyo Section IA, Math. 24(3), 629–644 (1977)MathSciNetMATHGoogle Scholar
  37. 37.
    Wang, M., Wu, Z.: Conoetherian coalgebras. Algebra Colloquium 5(1), 117–120 (1998)MathSciNetMATHGoogle Scholar
  38. 38.
    Yekutieli, A.: Dualizing complexes over noncommutative graded algebras. J. Algebra 153(1), 41–84 (1992)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Yekutieli, A., Zhang, J.J.: Rings with Auslander dualizing complexes. J. Algebra 213(1), 1–51 (1999). arXiv:9804005 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Natural SciencesUniversity of HaifaHaifaIsrael
  2. 2.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Sector of Algebra and Number TheoryInstitute for Information Transmission ProblemsMoscowRussia
  4. 4.Faculty of Mathematics and Physics, Department of AlgebraCharles UniversityPragueCzech Republic

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