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On jets, extensions and characteristic classes II


Author: Helge Maakestad
Journal: Proc. Amer. Math. Soc. 141 (2013), 151-169
MSC (2010): Primary 14F10, 14F40
DOI: https://doi.org/10.1090/S0002-9939-2012-11412-1
Published electronically: May 22, 2012
MathSciNet review: 2988719
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Abstract: In this paper we define the generalized Atiyah classes $ c_{\mathcal {J}}(\mathcal {E})$ and $ c_{\mathcal {O}_X}(\mathcal {E})$ of a quasi-coherent sheaf $ \mathcal {E}$ with respect to a pair $ (\mathcal {I},d)$, where $ \mathcal {I}$ is a left and right $ \mathcal {O}_X$-module and $ d$ a derivation. We relate this class to the structure of left and right modules on the first order jet bundle $ \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})$. In the main result of the paper we show $ c_{\mathcal {O}_X}(\mathcal {E})=0$ if and only if there is an isomorphism $ \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{left} \cong \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{right}$ as $ \mathcal {O}_X$-modules. We also give explicit examples where $ c_{\mathcal {O}_X}(\mathcal {E})\neq 0$ using jet bundles of line bundles on the projective line. Hence the classes $ c_{\mathcal {J}}(\mathcal {E})$ and $ c_{\mathcal {O}_X}(\mathcal {E})$ are nontrivial. The classes we introduce generalize the classical Atiyah class.


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  • 1. M. André, Homologie des algèbres commutatives, Grundlehren Math. Wiss. no. 206, Springer-Verlag (1974). MR 0352220 (50:4707)
  • 2. M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. no. 85 (1957). MR 0086359 (19:172c)
  • 3. A. Grothendieck, EGA IV. Étude locale de schémas et des morphismes de schémas, I, Publ. Math. IHES no. 20 (1964). MR 0173675 (30:3885)
  • 4. L. Illusie, Complexe cotangent et deformations I, Lecture Notes in Math. Vol. 239, Springer-Verlag (1971). MR 0491680 (58:10886a)
  • 5. L. Illusie, Complexe cotangent et deformations II, Lecture Notes in Math. Vol. 283, Springer-Verlag (1972). MR 0491681 (58:10886b)
  • 6. M. Karoubi, Homologie cyclique et $ K$-théorie, Astérisque no. 149 (1987). MR 913964 (89c:18019)
  • 7. H. Maakestad, A note on the principal parts on projective space and linear representations, Proc. Amer. Math. Soc., vol. 133, no. 2 (2005). MR 2093054 (2005h:14104)
  • 8. H. Maakestad, Chern classes and Lie-Rinehart algebras, Indagationes Math. no. 18 (2007). MR 2424316 (2009b:16022)
  • 9. H. Maakestad, Chern classes and $ \mathrm {Exan}$ functors, in progress (2009).
  • 10. H. Maakestad, Principal parts on the projective line over arbitrary rings, Manuscripta Math., vol. 126, no. 4 (2008). MR 2425435 (2009k:14034)
  • 11. H. Maakestad, On parameter spaces of right $ \mathcal {O}_X$-structures, in progress (2010).
  • 12. H. Maakestad, On the structure of jet bundles on projective space, in progress (2011).

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Additional Information

Helge Maakestad
Affiliation: Institut Fourier, 100 rue des maths, BP 74, 38402 St. Martin d’Hères cedex, France
Email: h{\textunderscore}maakestad@hotmail.com

DOI: https://doi.org/10.1090/S0002-9939-2012-11412-1
Keywords: Atiyah sequence, jet bundle, characteristic class, generalized Atiyah class, square zero extension, lifting
Received by editor(s): January 11, 2011
Received by editor(s) in revised form: January 28, 2011, April 12, 2011, and June 13, 2011
Published electronically: May 22, 2012
Communicated by: Lei Ni
Article copyright: © Copyright 2012 American Mathematical Society

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