About: Arithmetic and geometric Frobenius

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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 rp is a ring endomorphism of R. 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)

Attributes	Values
rdfs:label	Arithmetic and geometric Frobenius (en)
rdfs:comment	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 rp is a ring endomorphism of R. 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) (en)
dct:subject	Algebraic number theory Algebraic geometry Mathematical terminology
Wikipage page ID	6988121 (xsd:integer)
Wikipage revision ID	916378614 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime field Ring endomorphism Affine scheme Mathematics Subring Transport of structure Frobenius endomorphism Minus sign Commutative ring Spectrum of a ring Galois group Cyclic group Finite field Fibre product Prime number Surjective Algebraic number theory Algebraic geometry Mathematical terminology Characteristic (algebra) Grothendieck's relative point of view Scheme theory Inverse mapping Frobenius automorphism Springer-Verlag
sameAs	Arithmetic and geometric Frobenius Arithmetic and geometric Frobenius Arithmetic and geometric Frobenius
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Abstract-algebra-stub
has abstract	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 rp is a ring endomorphism of R. 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. (en)
gold:hypernym	Number
prov:wasDerivedFrom	wikipedia-en:Arithmetic_and_geometric_Frobenius?oldid=916378614&ns=0
page length (characters) of wiki page	2081 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Arithmetic_and_geometric_Frobenius
is Link from a Wikipage to another Wikipage of	Arithmetic and geometric frobenius Glossary of algebraic geometry Hasse–Witt matrix Grothendieck trace formula F-crystal List of things named after Ferdinand Georg Frobenius Frobenius mapping Arithmetic Frobenius Geometric Frobenius
is Wikipage redirect of	Arithmetic and geometric frobenius Frobenius mapping Arithmetic Frobenius Geometric Frobenius
is foaf:primaryTopic of	wikipedia-en:Arithmetic_and_geometric_Frobenius

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