Hochschild cohomology of cubic surfaces

详细信息    查看全文

• 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
• 参考文献：1. Alev, J., Farinati, M.A., Lambre, T., Solotar, A.L. (2000) Homologie des invariants d鈥檜ne alg猫bre de Weyl sous l鈥檃ction d鈥檜n groupe fini. Journal of Algebra, 232: pp. 564-577 f="http://dx.doi.org/10.1006/jabr.2000.8406" target="_blank" title="It opens in new window">CrossRef
  2. A. Brugui猫res, A. Cattaneo, B. Keller and C. Torossian, / D茅formation, Quantification, Th茅orie de Lie, Panoramas et Synth猫ses, SMF (2005).
  3. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. I and II. Physical applications, / Ann. Physics, 111 (1978), 61鈥?10 and 111鈥?51.
  4. C. Fronsdal and M. Kontsevich, Quantization on curves, / Lett. Math. Phys., 79 (2007), 109鈥?29, f="http://arxiv.org/abs/math-ph/0507021" class="a-plus-plus">math-ph/0507021.
  5. Gerstenhaber, M. (1963) The cohomology structure of an associative ring. Annals of Math. (2), 78: pp. 267-288 f="http://dx.doi.org/10.2307/1970343" target="_blank" title="It opens in new window">CrossRef
  6. L. Guieu and C. Roger, avec un appendice de Sergiescu V., L鈥橝lg猫bre et le Groupe de Virasoro: aspects g茅om茅triques et聽alg茅briques, g茅n茅ralisations, Publication du Centre de Recherches Math茅matiques de Montr茅al, s茅rie 鈥淢onographies, notes de cours et Actes de conf茅rences鈥? PM28 (2007).
  7. Hochschild, G., Kostant, B., Rosenberg, A. (1962) Differential forms on regular affine algebras. Trans. Amer. Math. Soc., 102: pp. 383-408 f="http://dx.doi.org/10.1090/S0002-9947-1962-0142598-8" target="_blank" title="It opens in new window">CrossRef
  8. M. Kontsevich, Deformation quantization of Poisson manifolds, I, Preprint IHES (1997), f="http://arxiv.org/abs/q-alg/9709040" class="a-plus-plus">q-alg/9709040.
  9. J. L. Loday, / Cyclic Homology, Springer-Verlag (Berlin, Heidelberg, 1998).
  10. A. Pichereau, Cohomologie de Poisson en dimension trois, / C.R. Acad. Sci. Paris, Ser. I, 340 (2005).
  11. Pichereau, A. (2006) Poisson (co)homology and isolated singularities. J. Algebra, 299: pp. 747-777 f="http://dx.doi.org/10.1016/j.jalgebra.2005.10.029" target="_blank" title="It opens in new window">CrossRef
  12. E. Rannou and P. Saux-Picart, / Cours de calcul formel, partie II, 茅ditions Ellipses (2002).
  13. Roger, C., Vanhaecke, P. (2002) Poisson cohomology of the affine plane. J. Algebra, 251: pp. 448-460 f="http://dx.doi.org/10.1006/jabr.2002.9147" target="_blank" title="It opens in new window">CrossRef
  14. B. Sturmfels, / Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation, Springer Verlag (Wien, New-York, 1993).
  15. M. Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, in: / Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), volume 8, (1994), pp. 213鈥?30.
  
• 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.