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 is a ring endomorphism of R.
The image of φ is then R, 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) → Spec(R)
of affine schemes. Even in cases where R = 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,/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) ※, vol. 13, "Berlin," New York: Springer-Verlag, ISBN 978-3-540-12175-6, MR 0926276, p. 5
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