Arithmetic and geometric Frobenius
Encyclopedia
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 field
s, 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
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 change
s, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois group
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
is defined in any commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R that has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p, where p is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. 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
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of R consisting of p-th powers. In some important cases, for example finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, 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
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
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 change
Grothendieck's relative point of view
Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object...
s, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s, or defined by transport of structure
Transport of structure
In mathematics, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists...
, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
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.