Skip to main content

Coring Categories and Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories


In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the relative group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over vec. We prove that the latter two groups are isomorphic.

This is a preview of subscription content, access via your institution.


  1. 1.

    Andruskiewitsch, N., Etingof, P., Gelaki, S.: Triangular Hopf algebras with the Chevalley property. Mich. Math. J. 49, 277–298 (2001)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bruguières, A.: Catégories prémodulaires, modularisations et invariants desvariétés de dimension 3. Math. Ann. 316, 215–236 (2000)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brzeziński, T.: The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebr. Represent. Theory 5, 389–410 (2002)

  4. 4.

    Caenepeel, S., Femić, B.: The Brauer group of Azumaya corings and the second cohomology group. K-Theory 34, 361–393 (2005)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Crane, L., Yetter, D.: Deformations of (Bi) tensor Categories. Cahiers Topologie Géom. Différentielle Catég 39(3), 163–180 (1998)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories I. Sel. Math. New Ser. 16(1), 1–119 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Davydov, A., Nikshych, D.: The Picard crossed module of a braided tensor category. Algebra Number Theory 7(6), 1365–1403 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dello, J., Zhang, Y.: Braided autoequivalences and the equivariant Brauer group of a quasi-triangular Hopf algebra. J. Algebra 445(1), 244–279 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Etingof, P., Gelaki, S.: The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field of characteristic 0. Mosc. Math. J. 3(1), 37–43 (2003)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. Lecture notes for MIT 18.769, 2009. (2009)

  11. 11.

    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Mathematical Surveys and Monographs, vol. 205. AMS, Providence (2015)

    Book  Google Scholar 

  12. 12.

    Etingof, P., Nikshych, D., Ostrik, V.: Fusion categories and homotopy theory. Quantum Topol. 1(3), 209–273 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Etingof, P., Ostrik, V.: Finite tensor categories. Mosc. Math. J. 4(3), 627–654 (2004)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Femić, B.: Villamayor–Zelinsky sequence for symmetric finite tensor categories. Appl. Categ. Struct. 25(6), 1199–1228 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Femić, B.: Villamayor–Zelinsky sequence for symmetric finite tensor categories. Updated preprint arXiv:1505.06504 [math.QA]

  16. 16.

    Femić, B.: Eilenberg–Watts theorem for 2-categories and quasi-monoidal structures for module categories over bialgebroid categories, accepted by. J. Pure Appl. Algebra 220(9), 3156–3181 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Fuchs, J., Schweigert, C., Valentino, A.: Bicategories for boundary conditions and for surface defects in 3-d TFT. Commun. Math. Phys. 321(2), 543–575 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Greenough, J.: Monoidal 2-structure of bimodule categories. J. Algebra 324, 1818–1859 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Greenough, J.: Relative centers and tensor products of tensor and braided fusion categories. J. Algebra 388, 374–396 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kitaev, A., Kong, L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313(2), 351–373 (2012)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Knus, M.A., Ojanguren, M.: Théorie de la descente et algèbres d’Azumaya. Lecture Notes in Mathematics, vol. 389. Springer, Berlin (1974)

    Book  Google Scholar 

  22. 22.

    Mombelli, M.: Una introducción a las categorías tensoriales y sus representaciones.

  23. 23.

    Müger, M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87(3), 291–308 (2003)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Neuchl, M.: Representation theory of Hopf categories, PhD thesis

  25. 25.

    Sweedler, M.E.: The predual theorem to the Jacobson–Bourbaki theorem. Trans. Am. Math. Soc. 213, 391–406 (1975)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Van Oystaeyen, F., Zhang, Y.H.: The Brauer group of a braided monoidal category. J. Algebra 202, 96–128 (1998)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Villamayor, O.E., Zelinsky, D.: Brauer groups and Amitsur cohomology for general commutative ring extensions. J. Pure Appl. Algebra 10, 19–55 (1977)

    MathSciNet  Article  Google Scholar 

Download references


This work was developed while the author worked at Facultad de Ingeniería, Universidad de la República in Montevideo, Uruguay. Author thanks to PEDECIBA and ANII in Uruguay.

Author information



Corresponding author

Correspondence to Bojana Femić.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Stephen Lack.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Femić, B. Coring Categories and Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories. Appl Categor Struct 29, 485–527 (2021).

Download citation


  • Brauer–Picard group
  • Finite tensor category
  • Symmetric monoidal category
  • Cohomology groups

Mathematics Subject Classification

  • 18D10
  • 16W30
  • 19D23