Nilpotent Elements in Hochschild Cohomology

  • Karin ErdmannEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)


We study the algebra \(A=K\langle x, y\rangle /(x^2, y^2, (xy)^k+q(yx)^k)\) over the field K where \(k\ge 1\) and where \(0\ne q \in K\). We determine a minimal projective bimodule resolution of A. In the case when q is not a root of unity, we compute its Hochschild cohomology. In particular, we show that for \(n\ge 3\), the nth part \(HH^n(A)\) has dimension \(k-1\) if char(K) does not divide k. We also show that every element in \(HH^n(A)\) for \(n\ge 1\) is nilpotent. This is motivated by the problem of understanding why the finite generation condition (Fg) fails, which is needed to ensure the existence of support varieties.


Algebra of dihedral type Socle deformation Bimodule resolution Hochschild cohomology Nilpotent elements in Hochschild cohomology 

MR Subject Classification

16E40 16S80 20C20 16G20 


  1. 1.
    D. J. Benson, Modular representation theory: new trends and methods. Lecture Notes in Mathematics, 1081 Springer-Verlag, Berlin, 1984.Google Scholar
  2. 2.
    D.J. Benson, Representation rings of finite groups, Representations of algebras (Durham, 1985), 181–199, London Math. Soc. Lecture Note Ser. 116, Cambridge Univ. Press, Cambridge, 1986.Google Scholar
  3. 3.
    D. J. Benson, Representations and cohomology I, Cambridge studies in advanced mathematics 30, 1991.Google Scholar
  4. 4.
    R. Buchweitz, E.L. Green, D. Madsen and Ø. Solberg, Finite Hochschild cohomology without finite global dimension. Math. Res. Lett. 12 (2005), no. 5–6, 805–816.MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Erdmann, Algebras with non-periodic bounded modules, to appear in J. Algebra 475 (2017), 308–326.MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. Erdmann, Blocks of tame representation type and related algebras, Springer Lecture Notes in Mathematics 1428 (1990).CrossRefGoogle Scholar
  7. 7.
    K. Erdmann, M. Holloway, N. Snashall, Ø. Solberg, R. Taillefer, Support varieties for selfinjective algebras, K-Theory, 33 (2004), 67–87.MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Happel, Hochschild cohomology of finite-dimensional algebras, Springer Lecture Notes in Mathematics 1404 (1989), 108–126.Google Scholar
  9. 9.
    T. Holm, Hochschild cohomology of tame blocks. J. Algebra 271 (2004)(2), 798–826.MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. M. Ringel, The indecomposable representations of the dihedral 2-groups. Math. Ann. 214 (1975), 19–34.MathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings. Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732.Google Scholar
  12. 12.
    Ø. Solberg, Support varieties for modules and complexes. Trends in representation theory of algebras and related topics, 239–270, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006.Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Mathematical Institute, University of OxfordOxfordUK

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