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Tate-Hochschild Cohomology of Radical Square Zero Algebras

  • Zhengfang Wang
Article
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Abstract

For algebras with radical square zero, we give a combinatorial description to the Tate-Hochschild cohomology. We compute the Gerstenhaber algebra structure on the Tate-Hochschild cohomology for some classes of such algebras.

Keywords

Radical square zero algebra Tate-Hochschild cohomology Gerstenhaber algebra BV algebra 

Mathematics Subject Classification (2010)

16E05 13D03 16G20 

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Notes

Acknowledgements

This work is a part of author’s PhD thesis. He would like to thank his supervisor Alexander Zimmermann for introducing this interesting topic and for his many valuable suggestions for improvement. He also would like to thank Huafeng Zhang for many useful discussions during this project. The author is indebted to Ragnar-Olaf Buchweitz for the constant support and encouragement.

The author is very grateful to the referee for valuable suggestions and comments. The author was partially supported by NSFC (No.11871071).

References

  1. 1.
    Abrams, G., Pino, G.A.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barot, M: Introduction to the Representation Theory of Algebras. Springer, Berlin (2015)CrossRefGoogle Scholar
  3. 3.
    Bergh, P.A., Jorgensen, D.A.: Tate-Hochschild homology and cohomology of Frobenius algebras. J. Noncommut. Geom. 7, 907–937 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Buchweitz, R.-O.: Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, manuscript. Universität Hannover (1986)Google Scholar
  5. 5.
    Chen, X.: The singularity category of an algebra with radical square zero. Doc. Math. 16, 921–936 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, X., Yang, D.: Homotopy categories, Leavitt path algebras and Gorenstein projective modules. Int. Math. Res. Not. IMRN 10, 2597–2633 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, X., Li, H., Wang, Z.: The Hochschild cohomology of Leavitt path algebras and Tate-Hochschild cohomology. To appearGoogle Scholar
  8. 8.
    Cibils, C.: Rigidity of truncated quiver algebras. Adv. Math. 79, 18–42 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cibils, C.: Rigid monomial algebras. Math. Ann. 289, 95–109 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cibils, C.: Hochschild cohomology algebra of radical square zero algebras, Algebras and Modules II (Geiranger, 1996). In: CMS Conference Proceedings, 24, pp 93–101. American Mathematical Society, Providence (1998)Google Scholar
  11. 11.
    Cibils, C., Solotar, A.: Hochschild cohomology algebra of abelian groups. Arch. Math. 68, 17–21 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cuntz, J., Quillen, D.: Extensions and nonsingularity. J. Am. Math. Soc. 8, 251–289 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Eu, C.-H, Schedler, T.: Calabi-Yau Frobenius algebras. J. Algebra 321(3), 774–815 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Keller, B: Singular Hochschild cohomology via the singularity category, arXiv:1809.05121
  16. 16.
    Loday, J.-L., Vallette, B.: Algebraic Operads, Grundlehren der mathematischen Wissenschaften, 346. Springer, Berlin (2012)zbMATHGoogle Scholar
  17. 17.
    Markl, M.: Operads and prop’s, in Elsevier. Handb. Algebra 5, 87–140 (2008)Google Scholar
  18. 18.
    Nguyen, V.C.: Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras. J. Pure Appl. Algebra 217, 1967–1979 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Volume II, 503531, Progress in Mathematics, 270. Birkhuser Boston, Inc, Boston (2009)Google Scholar
  20. 20.
    Rivera, M., Wang, Z.: Singular Hochschild cohomology and algebraic string operations. Accepted for publication in Journal of Noncommutative Geometry. arXiv:1703.03899
  21. 21.
    Sánchez-Flores, S.: The Lie structure on the Hochschild cohomology groups of monomial algebras with radical square zero. J. Algebra 320, 4249–4269 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sánchez-Flores, S.: The Lie structure on the Hochschild cohomology d’algèbres monomiales, Mathematics. Université Montpellier II - Sciences et Techniques du Languedoc (2009)Google Scholar
  23. 23.
    Smith, S.P.: Category equivalences involving graded modules over path algebras of quivers. Adv. Math. 230, 1780–1810 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, Z: Gerstenhaber algebra and Deligne’s conjecture on Tate-Hochschild cohomology, arXiv:1801.07990
  25. 25.
    Weibel, C.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  26. 26.
    Zimmermann, A.: Representation Theory: A Homological Algebra Point of View. Springer, London (2014)zbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  2. 2.Université Paris Diderot-Paris 7, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS UMR 7586Paris Cedex 13France

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