Abstract
This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.
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References
Auslander, M., Reiten, I.: Cohen–Macaulay and Gorenstein Artin algebras. In: Representation Theory of Finite Groups and Finite-Dimensional Algebras. Birkhäuser, Basel, pp. 221–245 (1991)
Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge University Press, Cambridge (1995)
Brown, K.: Cohomology of Groups. Springer, New York (1982)
Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings. http://hdl.handle.net/1807/16682 (1986)
Carlson, J.F.: Periodic modules over modular group algebras. J. Lond. Math. Soc. (2) 15(3), 431–436 (1977)
Dotsenko, V., Gélinas, V., Tamaroff, P.: Finite generation for Hochschildcohomology of Gorenstein monomial algebras. arXiv:1909.00487 (2019)
Erdmann, K., Skowroński, A.: Periodic algebras. In: Trends in Representation Theory of Algebras and Related Topics. European Mathematical Society, Zürich, pp. 201–251 (2008)
Erdmann, K., Skowroński, A.: Algebras of generalized quaternion type. Adv. Math. 349, 1036–1116 (2019)
Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras I. J. Algebra 205(2), 391–412 (1998)
Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras II. J. Algebra 205(2), 413–434 (1998)
Eu, C.-H.: The product in the Hochschild cohomology ring of preprojective algebras of Dynkin quivers. J. Algebra 320(4), 1477–1530 (2008)
Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. 78(2), 267–288 (1963)
Green, E.L., Snashall, N., Solberg, Ø.: The Hochschild cohomology ring of a selfinjective algebra of finite representation type. Proc. Amer. Math. Soc. 131(11), 3387–3393 (2003)
Liu, Y., Zhou, G., Zimmermann, A.: Higman ideal, stable Hochschild homology and Auslander–Reiten conjecture. Math. Z. 270(3–4), 759–781 (2012)
Lu, M., Zhu, B.: Singularity categories of Gorenstein monomial algebras. J. Pure Appl. Algebra 225(8), 106651 (2021)
Nguyen, V.C.: The Tate–Hochschild cohomology ring of a group algebra. arXiv:1212.0774 (2012)
Wang, Z.: Singular equivalence of Morita type with level. J. Algebra 439, 245–269 (2015)
Wang, Z.: Gerstenhaber algebra and Deligne’s conjecture on Tate–Hochschild cohomology. arXiv:1801.07990 (2018)
Wang, Z.: Invariance of the Gerstenhaber algebra structure on Tate–Hochschild cohomology. J. Inst. Math. Jussieu (2019). https://doi.org/10.1017/S1474748019000367
Wang, Z.: Tate–Hochschild cohomology of radical square zero algebras. Algebras Represent. Theory 23(1), 169–192 (2020)
Zimmermann, A.: Representation Theory. A Homological Algebra Point of View. Springer, Cham (2014)
Acknowledgements
The author would like to thank the referee for a careful reading of the paper and for helpful suggestions. The author also would like to thank Professor Katsunori Sanada for valuable advice. The author is grateful to Professor Tomohiro Itagaki and Professor Ayako Itaba for constructive comments.
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Usui, S. Tate–Hochschild cohomology for periodic algebras. Arch. Math. 116, 647–657 (2021). https://doi.org/10.1007/s00013-021-01576-2
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DOI: https://doi.org/10.1007/s00013-021-01576-2