# 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.

## References

- Freitag, Eberhard; Kiehl, Reinhardt (1988),
*Étale cohomology and the Weil conjecture*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],**13**, Berlin, New York: Springer-Verlag, ISBN 978-3-540-12175-6, MR 926276, p. 5