Arithmetic and geometric Frobenius

Arithmetic and geometric Frobenius

In mathematics, the Frobenius endomorphism is defined in any commutative ring "R" that has characteristic "p", where "p" is a prime number. Namely, the mapping φ that takes "r" in "R" to "r""p" is a ring endomorphism of "R".

The image of φ is then "R""p", the subring of "R" consisting of "p"-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring "automorphism".

The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping

:φ*: Spec("R""p") → Spec("R")

of affine schemes. Even in cases where "R""p" = "R" this is not the identity, unless "R" is the prime field.

Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called "geometric Frobenius". The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.


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