Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-29T13:24:58.906Z Has data issue: false hasContentIssue false
  • English
  • Français

INVARIANCE OF THE GERSTENHABER ALGEBRA STRUCTURE ON TATE-HOCHSCHILD COHOMOLOGY

Part of: Homological methods Modules, bimodules and ideals Homological algebra Abelian categories

Published online by Cambridge University Press:  15 July 2019

Zhengfang Wang
Zhengfang Wang*
Affiliation:
Beijing International Center for Mathematical Research (BICMR), Peking University, No. 5 Yiheyuan Road Haidian District, Beijing100871, P.R. China Université Paris Diderot-Paris 7, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate–Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.

Keywords

Gerstenhaber algebraSingularity categoryTate–Hochschild cohomology

MSC classification

Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 18E30: Derived categories, triangulated categories 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 18G10: Resolutions; derived functors
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 893 - 928
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernstein, J. and Lunts, V., Equivariant Sheaves and Functors, Lecture Notes in Mathematics, Volume 1578 (Springer, Berlin, 1994).CrossRefGoogle Scholar
Buchweitz, R.-O., Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings, Universität Hannover (1986), available at http://hdl.handle.net/1807/16682.Google Scholar
Deligne, P., SGA $4\frac{1}{2}$ -Cohomologie étale, Lecture Notes in Mathematics, Volume 569 (Springer-Verlag, Berlin, 1977).Google Scholar
Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of Math. (2) 78(2) (1963), 267–288.CrossRefGoogle Scholar
Keller, B., Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), 223–273.CrossRefGoogle Scholar
Keller, B., Hochschild cohomology and derived Picard groups, J. Pure Appl. Algebra 136 (1999), 1–56.CrossRefGoogle Scholar
Keller, B., Singular Hochschild cohomology via the singularity category, C. R. Acad. Sci. Math. 356(11–12) (2018), 1106–1111.CrossRefGoogle Scholar
Keller, B. and Vossieck, D., Sous les catégories dérivées, C. R. Acad. Sci. 305(6) (1987), 225–228.Google Scholar
Koenig, S., Liu, Y. and Zhou, G., Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture, Trans. Amer. Math. Soc. 364(1) (2012), 195–232.CrossRefGoogle Scholar
Loday, J.-L., Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, 301, (Springer-Verlag, Berlin, 1992).CrossRefGoogle Scholar
Orlov, D., Triangulated Categories of Singularities and D-branes in Landau–Ginzburg Models, Trudy Matematicheskogo Instituta Steklova, 246, pp. 240–262 (Algebr. Geom. Metody, Svyazi i Prilozh, 2004).Google Scholar
Rickard, J., Morita theory for derived categories, J. Lond. Math. Soc. 39(2) (1989), 303–317.Google Scholar
Stasheff, J., The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra 89 (1993), 231–235.CrossRefGoogle Scholar
Wang, Z., Singular Hochschild cohomology and Gerstenhaber algebra structure, Preprint, 2015, arXiv:1508.00190.Google Scholar
Wang, Z., Singular equivalence of Morita type with level, J. Algebra 439 (2015), 245–269.CrossRefGoogle Scholar
Wang, Z., Gerstenhaber algebra and Deligne’s conjecture on Tate–Hochschild cohomology, Preprint, 2018, arXiv:1801.07990. To appear in Trans. Amer. Math. Soc.Google Scholar
Weibel, C., An Introduction to Homological Algebra (Cambridge University Press, Cambridge, 1995).Google Scholar
Zimmermann, Alexander, Representation Theory: A Homological Algebra Point of View (Springer, London, 2014).CrossRefGoogle Scholar