Mackey Algebras which are Gorenstein

  • Ivo Dell’Ambrogio
  • Jan ŠťovíčekEmail author


We correct and complete the picture available in the literature by showing that the integral Mackey algebra is Gorenstein if and only if the group order is square-free, in which case it must have Gorenstein dimension one.


Mackey functors Gorenstein rings Burnside rings 

Mathematics Subject Classification (2010)

20C20 16E10 (Primary) 19A22 16E65 (Secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank Serge Bouc, John Greenlees, Radu Stancu and Peter Symonds for useful discussions.


  1. 1.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bruns, Winfried, Jürgen, Herzog.: Cohen-Macaulay Rings, volume 39 of Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge (1993)Google Scholar
  3. 3.
    Bouc, S., Stancu, R., Webb, P.: On the projective dimensions of Mackey functors. Algebr. Represent. Theor. 20(6), 1467–1481 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dell’Ambrogio, I., Stevenson, G., Šťovíček, J.: Gorenstein homological algebra and universal coefficient theorems. Math. Z. 287(3–4), 1109–1155 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Greenlees, J.P.C.: Some remarks on projective Mackey functors. J. Pure Appl. Algebra 81(1), 17–38 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gustafson, W.H.: Burnside rings which are Gorenstein. Comm. Algebra 5(1), 1–15 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hall, M. Jr: The Theory of Groups. The Macmillan Co., New York (1959)zbMATHGoogle Scholar
  8. 8.
    Krämer, H.: Über die injektive Dimension des Burnsideringes einer endlichen Gruppe. J. Algebra 30, 294–304 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lindner, H.: A remark on Mackey-functors. Manuscripta Math. 18(3), 273–278 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rees, D.: A theorem of homological algebra. Proc. Cambridge Philos. Soc. 52, 605–610 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rognerud, B.: Trace maps for Mackey algebras. J. Algebra 426, 288–312 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Thévenaz, J., Webb, P.: The structure of Mackey functors. Trans. Amer. Math. Soc. 347(6), 1865–1961 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Paul PainlevéUniversité de Lille 1Villeneuve-d’Ascq CedexFrance
  2. 2.Department of AlgebraCharles University in PraguePraha 8Czech Republic

Personalised recommendations