In mathematics, a Frobenius splitting, introduced by txt, is a splitting of the injective morphism OX→F*OX from a structure sheaf OX of a characteristic p > 0 variety X to its image F*OX under the Frobenius endomorphism F*.

Brion & Kumar (2005) give a detailed discussion of Frobenius splittings.

A fundamental property of Frobenius-split projective schemes X is that the higher cohomology Hi(X,L) (i > 0) of ample line bundles L vanishes.

References

  • Brion, Michel & Kumar, Shrawan (2005), Frobenius splitting methods in geometry and representation theory, Vol. 231, Progress in Mathematics, Boston, MA: Birkhäuser Boston, doi:10.1007/b137486. ISBN 978-0-8176-4191-7. MR 2107324
  • Mehta, V. B. & Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for Schubert varieties", Annals of Mathematics. 122 (1): 27–40, Second Series, doi:10.2307/1971368. ISSN 0003-486X. JSTOR 1971368. MR 799251
  • Conference on Frobenius splitting in algebraic geometry, commutative algebra, and representation theory at Michigan, 2010.