Bundle of principal parts

In algebraic geometry, given a line bundle L on a smooth variety X, the bundle of n-th order principal parts of L is a vector bundle of rank that, roughly, parametrizes n-th order Taylor expansions of sections of L.

Precisely, let I be the ideal sheaf defining the diagonal embedding and the restrictions of projections to . Then the bundle of n-th order principal parts is[1]

Then and there is a natural exact sequence of vector bundles[2]

where is the sheaf of differential one-forms on X.

See also

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References

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  1. ^ Fulton 1998, Example 2.5.6.
  2. ^ SGA 6 1971, Exp II, Appendix II 1.2.4.
  • Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
  • Appendix II of Exp II of Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.