Hochschild cohomology of cubic surfaces

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• 作者：F. Butin (1)
  
  1. Institut Camille Jordan ; Universit茅 de Lyon ; Universit茅 Lyon 1 ; CNRS ; UMR5208 ; 43 blvd du 11 novembre 1918 ; F-69622 ; Villeurbanne-Cedex ; France
  
• 关键词：Hochschild cohomology ; Hochschild homology ; cubic surface ; Groebner basis ; algebraic resolution ; quantization ; star ; product ; 53D55 ; 13P10 ; 13D03
• 刊名：Acta Mathematica Hungarica
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：145
• 期：2
• 页码：263-282
• 全文大小：358 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2632

文摘

We consider the polynomial algebra \({\mathbb{C}[{\bf z}] := \mathbb{C}[z_1, z_2, z_3]}\) and the polynomial \({f := z^{3}_{1} + z^3_2 + z^3_3 + 3qz_1z_2z_3}\) , where \({q \in \mathbb{C}}\) . Our aim is to compute the Hochschild homology and cohomology of the cubic surface \({\mathcal{X}_f := \{{\bf z} \in \mathbb{C}^3/f({\bf z}) = 0\}}\) . For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology and cohomology by means of multivariate division and Groebner bases. Some formal computations with Maple are also used.