Frobenius splitting

In mathematics, a Frobenius splitting, introduced by Mehta and Ramanathan (1985), is a splitting of the injective morphism O_X→F_*O_X from a structure sheaf O_X of a characteristic p > 0 variety X to its image F_*O_X 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, Progress in Mathematics, vol. 231, 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, Second Series, 122 (1): 27–40, doi:10.2307/1971368, ISSN 0003-486X, JSTOR 1971368, MR 0799251

Conference on Frobenius splitting in algebraic geometry, commutative algebra, and representation theory at Michigan, 2010.

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