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 ÃÂ that takes r in R to r<sup>p</sup> is a ring endomorphism of R.
The image of ÃÂ is then R<sup>p</sup>, the subring of R consisting of p-th powers. In some important cases, for example finite fields, ÃÂ is surjective. Otherwise ÃÂ is an endomorphism but not a ring automorphism.
The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to ÃÂ. This gives a mapping
of affine schemes. Even in cases where R<sup>p</sup> = R this is not the identity, unless R is the prime field.
Mappings created by fibre product with ÃÂ*, 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.