Notes on Formal Smoothness

  • Tomasz Brzeziński
Part of the Trends in Mathematics book series (TM)


The definition of an S-category is proposed by weakening the axioms of a Q-category introduced by Kontsevich and Rosenberg. Examples of Q- and S-categories and (co)smooth objects in such categories are given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Ardizzoni, Separable functors and formal smoothness, J. K-Theory, in press (arXiv:math.QA/0407095).Google Scholar
  2. [2]
    A. Ardizzoni, T. Brzeziński and C. Menini, Formally smooth bimodules, J. Pure Appl. Algebra 212 (2008) 1072–1085.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Ardizzoni, C. Menini and D. Ştefan, Hochschild cohomology and’ smoothness’ in monoidal categories, J. Pure Appl. Algebra 208 (2007), 297–330.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    T. Brzeziński, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory 5 (2002), 389–410.CrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Brzeziński and R. Wisbauer, Corings and Comodules, Cambridge University Press, Cambridge, 2003.Google Scholar
  6. [6]
    S. Caenepeel, E. De Groot and J. Vercruysse, Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties, Trans. Amer. Math. Soc. 359 (2007), 185–226.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    S. Caenepeel, G. Militaru and S. Zhu, Frobenius and Separable Functors for Generalized Hopf Modules and Nonlinear Equations, Springer, Berlin (2002)Google Scholar
  8. [8]
    J. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer.Math. Soc. 8 (1995), 251–289.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    L. El Kaoutit and J. Gómez-Torrecillas, Comatrix corings: Galois corings, descent theory, and a structure theorem for cosemisimple corings, Math. Z. 244 (2003), 887–906.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    F. Guzman, Cointegrations, relative cohomology for comodules and coseparable corings, J. Algebra 126 (1989), 211–224.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P.J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer, New York, (1971).zbMATHGoogle Scholar
  12. [12]
    M. Kontsevich and A. Rosenberg, Noncommutative spaces, Preprint MPIM2004-35, 2004.Google Scholar
  13. [13]
    C. NĂstĂsescu, S. Raianu and F. Van Oystaeyen, Modules graded by G-sets, Math. Z. 203 (1990), 605–627.CrossRefMathSciNetGoogle Scholar
  14. [14]
    C. NĂstĂsescu, M. Van den Bergh and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397–413.CrossRefMathSciNetGoogle Scholar
  15. [15]
    M.D. Rafael, Separable functors revisited, Comm. Algebra 18 (1990), 1445–1459.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    W.F. Schelter, Smooth algebras, J. Algebra 103 (1986), 677–685.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, PA, 1991.zbMATHGoogle Scholar
  18. [18]
    R. Wisbauer, On Galois comodules, Comm. Algebra 34 (2006), 2683–2711.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Zarouali Darkaoui, Adjoint and Frobenius Pairs of Functors, Equivalences and the Picard Group for Corings, PhD Thesis, University of Granada, 2007.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Tomasz Brzeziński
    • 1
  1. 1.Department of MathematicsSwansea UniversitySwanseaUK

Personalised recommendations