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Finite generation of cohomology for Drinfeld doubles of finite group schemes

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Abstract

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual to \({{\,\mathrm{rep}\,}}(G)\) are also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of Friedlander and the author.

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References

  1. Arkhipov, S., Bezrukavnikov, R., Ginzburg, V.: Quantum groups, the loop Grassmannian, and the Springer resolution. J. Am. Math. Soc. 17(3), 595–678 (2004)

    Article  MathSciNet  Google Scholar 

  2. Avramov, L.L., Sun, L.-C.: Cohomology operators defined by a deformation. J. Algebra 204(2), 684–710 (1998)

    Article  MathSciNet  Google Scholar 

  3. Benson, D., Iyengar, S. B., Krause, H., Pevtsova, J.: Detecting nilpotence and projectivity over finite unipotent supergroup schemes. Preprint arXiv:1901.08273

  4. Bezrukavnikov, R., Ginzburg, V.: On deformations of associative algebras. Ann. Math. 166, 533–548 (2007)

    Article  MathSciNet  Google Scholar 

  5. Ciocan-Fontanine, I., Kapranov, M.: Derived quot schemes. Ann. Sci. Éc. Norm. Supér. 34(3), 403–440 (2001)

    Article  MathSciNet  Google Scholar 

  6. Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)

    Article  MathSciNet  Google Scholar 

  7. Drupieski, C.M.: Cohomological finite-generation for finite supergroup schemes. Adv. Math 288, 1360–1432 (2016)

    Article  MathSciNet  Google Scholar 

  8. Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)

    Article  MathSciNet  Google Scholar 

  9. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories, vol. 205. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Friedlander, E.M., Negron, C.: Cohomology for Drinfeld doubles of some infinitesimal group schemes. Algebra Number Theory 12(5), 1281–1309 (2018)

    Article  MathSciNet  Google Scholar 

  12. Friedlander, E.M., Parshall, B.J.: Cohomology of Lie algebras and algebraic groups. Am. J. Math. 108(1), 235–253 (1986)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Friedlander, E.M., Suslin, A.: Cohomology of finite group schemes over a field. Invent. Math. 127(2), 209–270 (1997)

    Article  MathSciNet  Google Scholar 

  15. Gelaki, S.: Module categories over affine group schemes. Quant. Topol. 6(1), 1–37 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Gulliksen, T.H.: A change of ring theorem with applications to Poincaré series and intersection multiplicity. Math. Scand. 34(2), 167–183 (1974)

    Article  MathSciNet  Google Scholar 

  17. Milne, J.S.: Algebraic Groups: The Theory of Group Schemes of Finite Type Over a Field, vol. 170. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

  18. Müger, M.: From subfactors to categories and topology II: the quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)

    Article  MathSciNet  Google Scholar 

  19. Negron, C., Pevtsova, J.: Hypersurface support and prime ideal spectra for stable categories. arXiv:2101.00141

  20. Negron, C., Pevtsova, J.: Support for integrable Hopf algebras via noncommutative hypersurfaces. arXiv:2005.02965

  21. Negron, C., Plavnik, J.: Cohomology of finite tensor categories: duality and Drinfeld centers. arXiv:1807.08854

  22. Sanada, K.: On the Hochschild cohomology of crossed products. Commun. Algebra 21(8), 2727–2748 (1993)

    Article  MathSciNet  Google Scholar 

  23. Suslin, A., Friedlander, E.M., Bendel, C.P.: Support varieties for infinitesimal group schemes. J. Am. Math. Soc. 10(3), 729–759 (1997)

    Article  MathSciNet  Google Scholar 

  24. Tabuada, G.: Une structure de catégorie de modeles de Quillen sur la catégorie des dg-catégories. Comptes Rendus Mathematique 340(1), 15–19 (2005)

    Article  MathSciNet  Google Scholar 

  25. Toén, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)

    Article  MathSciNet  Google Scholar 

  26. Touzé, A., van der Kallen, W.: Bifunctor cohomology and cohomological finite generation for reductive groups. Duke Math. J. 151(2), 251–278 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Thanks to Ben Briggs, Christopher Drupieski, Eric Friedlander, Julia Pevtsova, Antoine Touzé, and Sarah Witherspoon for helpful conversations. The proofs of Lemmas 2.3 and 2.4 are due to Ben Briggs and Ragnar Buchweitz (with any errors in their reproduction due to myself). I thank the referee for thoughtful comments which helped improve the text. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester.

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Cris Negron

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Negron, C. Finite generation of cohomology for Drinfeld doubles of finite group schemes. Sel. Math. New Ser. 27, 26 (2021). https://doi.org/10.1007/s00029-021-00637-2

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