In mathematics, the Frobenius endomorphism is defined in any commutative ring R {\displaystyle R} that has characteristic p {\displaystyle p} , where p {\displaystyle p} is a prime number. Namely, the mapping ϕ : r ↦ r p {\displaystyle \phi :r\mapsto r^{p}} is a ring endomorphism of R {\displaystyle R} . The image of ϕ {\displaystyle \phi } is then R p {\displaystyle R^{p}} , the subring of R {\displaystyle R} consisting of p {\displaystyle p} -th powers. In some important cases, for example finite fields, ϕ {\displaystyle \phi } is surjective. Otherwise, ϕ {\displaystyle \phi } is an endomorphism but not a ring automorphism. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to ϕ {\displaystyle \phi } . This gives a mapping
ϕ ∗ : Spec ( R p ) → Spec ( R ) {\displaystyle \phi ^{*}:\operatorname {Spec} (R^{p})\to \operatorname {Spec} (R)}
of affine schemes. Even in cases where R p = R {\displaystyle R^{p}=R} this is not the identity, unless R {\displaystyle R} is the prime field. Mappings created by fibre product with ϕ ∗ {\displaystyle \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.