In mathematics, a Frobenius splitting, introduced by , 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_*.
The concept is named after Ferdinand Georg Frobenius

Ferdinand Georg Frobenius

Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...

.

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 bundle

Ample line bundle

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space. An ample line bundle is one such that some positive power is very ample...

s L vanishes.

External links

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