Spectra of Tensor Triangulated Categories over a Category Algebra—A New Approach


We provide a new method to compute the Balmer spectra of the bounded derived category and the singularity category of the category algebra of a finite EI category by a decomposition trick due to Stevenson. In particular, we reobtain the result on the singularity category given by Wang under a weaker condition.

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

Data Availability Statement

My manuscript has no associated data.


  1. 1.

    Baland, S., Chirvasitu, A., Stevenson, G.: The prime spectra of relative stable module categories. Tran. Amer. Math. Soc. 371(1), 489–503 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine. Math. 588, 149–168 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Balmer, P.: Tensor triangular geometry. In Proc. Inter. Cong. Math., vol. 2, pp. 85–112. Hindustan Book Agency New Delhi (2010)

  4. 4.

    Balmer, P., Sanders, B.: The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math. 208(1), 283–326 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Barthel, T., Hausmann, M., Naumann, N., Nikolaus, T., Noel, J., Stapleton, N.: The Balmer spectrum of the equivariant homotopy category of a finite abelian group. Invent. Math. 216(1), 215–240 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Benson, D., Iyengar, S., Krause, H.: Module categories for group algebras over commutative rings. J. K-Theory 11(2), 297–329 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dubey, U.V., Mallick, V.M.: Spectrum of some triangulated categories. J. Alg. 364, 90–118 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2016)

    MATH  Google Scholar 

  9. 9.

    Friedlander, E.M., Pevtsova, J.: \(\pi \)-supports for modules for finite group schemes. Duke Math. J. 139(2), 317–368 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hügel, L.A., Koenig, S., Liu, Q.H., Yang, D.: Ladders and simplicity of derived module categories. J. Alg. 472, 15–66 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Liu, P., Lu, M.: Recollements of singularity categories and monomorphism categories. Commun. Algebra 43(6), 2443–2456 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Rotman, J.J.: An Introduction to Homological Algebra Universitext. Springer, Berlin (2008)

    Google Scholar 

  13. 13.

    Wang, R.: Gorenstein triangular matrix rings and category algebras. J. Pure Appl. Alg. 220(2), 666–682 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wang, R.: The spectrum of the singularity category of a category algebra. Appl. Categ. Struct. 27(4), 427–433 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Webb, P.: An Introduction to the Representations and Cohomology of Categories, Group represent. theory, pp. 149-173. EPFL Press, Lausanne (2007)

  16. 16.

    Weibel, C.A.: An Introduction to Homological Algebra, vol. 38. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  17. 17.

    Xu, F.: Tensor structure on \({k{\cal{C}}}\)-mod and cohomology. Proc. Edin. Math. Soc. 56(1), 349–370 (2013)

    Article  Google Scholar 

  18. 18.

    Xu, F.: Spectra of tensor triangulated categories over category algebras. Arch. Math. 103(3), 235–253 (2014)

    MathSciNet  Article  Google Scholar 

Download references


I would like to thank professor Changchang Xi for introducing me to the theory of tensor triangular geometry. I like to thank professor Hongxing Chen and the anonymous referee who carefully read my work and provided insightful comments which helped me improve the paper.

Author information



Corresponding author

Correspondence to Yaohua Zhang.

Additional information

Publisher's Note

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

Communicated by Henning Krause.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y. Spectra of Tensor Triangulated Categories over a Category Algebra—A New Approach. Appl Categor Struct (2021). https://doi.org/10.1007/s10485-021-09648-8

Download citation


  • Tensor triangulated category
  • Triangular spectrum
  • Singularity category
  • Semi-orthogonal decomposition
  • Finite EI category
  • Category algebra

Mathematics Subject Classification

  • Primary: 18D10
  • 18E30
  • Secondary: 16D90
  • 16E35
  • 16G10