In mathematics, a Frobenius splitting, introduced by , is a splitting of the

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

O_''X''→F_*O_''X'' from a

structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...

O_''X'' of a characteristic ''p'' > 0 variety ''X'' to its image F_*O_''X'' under the

Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...

F_*. give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split

projective scheme In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

s ''X'' is that the

higher cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be vie ...

''H''^''i''(''X'',''L'') (''i'' > 0) of

ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...

s ''L'' vanishes.

References

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External links

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