The image of φ is then Rp, 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(Rp) → Spec(R)
of affine schemes. Even in cases where Rp = 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.
Referencesedit
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)], vol. 13, Berlin, New York: Springer-Verlag, ISBN978-3-540-12175-6, MR 0926276, p. 5
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arithmetic, geometric, frobenius, mathematics, frobenius, endomorphism, defined, commutative, ring, that, characteristic, where, prime, number, namely, mapping, that, takes, ring, endomorphism, image, then, subring, consisting, powers, some, important, cases, . 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 f that takes r in R to rp is a ring endomorphism of R The image of f is then Rp the subring of R consisting of p th powers In some important cases for example finite fields f is surjective Otherwise f is an endomorphism but not a ring automorphism The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to f This gives a mapping f Spec Rp Spec R of affine schemes Even in cases where Rp R this is not the identity unless R is the prime field Mappings created by fibre product with f 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 editFreitag Eberhard Kiehl Reinhardt 1988 Etale cohomology and the Weil conjecture Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Results in Mathematics and Related Areas 3 vol 13 Berlin New York Springer Verlag ISBN 978 3 540 12175 6 MR 0926276 p 5 nbsp This algebraic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Arithmetic and geometric Frobenius amp oldid 1170034935, wikipedia, wiki, book, books, library,