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 $\phi :r\mapsto r^{p}$ is a ring endomorphism of $R$.

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

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

$\phi ^{*}:\operatorname {Spec} (R^{p})\to \operatorname {Spec} (R)$

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 $\phi ^{*}$, 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. A Series of Modern Surveys in Mathematics. Vol. 13. Berlin, Heidelberg: Springer. ISBN 978-3-540-12175-6. MR 0926276.