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Hochschild cohomology of Beilinson algebra of exterior algebra

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Abstract

Let Λ n be the Beilinson algebra of exterior algebra of an n-dimensional vector space, which is derived equivalent to the endomorphism algebra \(End_{\mathcal{O}_X } (T)\) of a tilting complex T = Π n i=0 O X (i) of coherent O X -modules over a projective scheme X = P n k . In this paper we first construct a minimal projective bimodule resolution of Λ n , and then apply it to calculate k-dimensions of the Hochschild cohomology groups of Λ n in terms of parallel paths. Finally, we give an explicit description of the cup product and obtain a Gabriel presentation of Hochschild cohomology ring of Λ n . As a consequence, we provide a class of algebras of finite global dimension whose Hochschild cohomology rings have non-trivial multiplicative structures.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Hubei University, Wuhan, 430062, China

    YunGe Xu, Chao Zhang, XiaoJing Ma & QingFeng Hu

  2. Academy of Mathematics and System Science, Chinese Academy of Science, Beijing, 100190, China

    Chao Zhang

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  1. YunGe Xu
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  2. Chao Zhang
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  3. XiaoJing Ma
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  4. QingFeng Hu
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Correspondence to YunGe Xu.

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Xu, Y., Zhang, C., Ma, X. et al. Hochschild cohomology of Beilinson algebra of exterior algebra. Sci. China Math. 55, 1153–1170 (2012). https://doi.org/10.1007/s11425-012-4388-9

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