Frobenius endomorphism

The Frobeniushomomorphismus is in algebra is an endomorphism of rings, whose characteristic is a prime number. The Frobeniushomomorphismus is named after the German mathematician Ferdinand Georg Frobenius.

• 3.1 Definition
• 3.2 Example
• 3.3 Features
• 3.4 set of long-
• 3.5 Frobenius and shift for commutative groups

Frobeniusendomorphismus a ring

Definition

It is a commutative ring unitary with the characteristic, wherein a prime number. As Frobeniushomomorphismus is the picture

Referred to. It is a ring homomorphism.

If, then also

A ring homomorphism.

Proof of Homomorphieeigenschaft

The figure is compatible with the multiplication in, since due to the power laws

Applies. Also applies Interestingly, the figure also with the addition in compatible, which means it applies. With the help of Binomialsatzes namely follows

Since a prime number, namely but not for shares. Therefore, since the characteristic of the counter but not the denominator of the binomial coefficient

Shares, the binomial coefficients vanish in the above formula. The addition simplifies to

And is compatible with the addition in. This relationship is referred to as Freshmen 's Dream, the dream of the beginner, at the English-speaking world.

Use

Below is always a prime number and a power of. Disclaimer: All rings have characteristic.

• After the little theorem of Fermat is the identity on the residue class ring. General: Is a finite field, then the identity.
• A body, then.
• Is an extension finite field, then an automorphism of, the element can be determined. The Galois group is cyclic and generated by.
• Is a ring, then iff is injective if no nontrivial nilpotent elements. ( The core of. )
• Is a ring and is bijective, then that means the ring perfect ( or perfect). In a perfect ring, every element has a unique nth root. Perfect bodies are characterized by the fact that they have no inseparable extensions.
• The perfect end to a ring can be represented as the inductive limit:

• The additivity of the mapping is also exploited in the Artin - Schreier theory.

Frobeniusautomorphismen of local and global bodies

The following assumptions are used to describe both the case of a finite Galois extension of algebraic number fields and local bodies. Be a Dedekind ring, its quotient field, a finite Galois extension, the whole concluding in. Then is a Dedekind ring. Let further a maximal ideal in with finite residue field, and also. The field extension is Galois. Be the Galois group of. It operates transitively on the set of prime ideals lying. Be the decomposition group, that is, the stabilizer of. The induced homomorphism

Is surjective. Its core is the inertia group.

It is now unbranched, that is,. Then the homomorphism is an isomorphism. The Frobenius (also Frobeniuselement ) is the archetype of the Frobenius below. He is uniquely characterized by the following property:

Because operates on the prime ideals over transitive, the Frobeniusautomorphismen are conjugated to them, so that their conjugacy class is defined by unique. If the extension is abelian, we obtain a unique Frobenius.

Frobeniusautomorphismen are of central importance for the class field theory: In the ideal theoretical formulation of the Reziprozitätsabbildung is induced by the mapping. Conjugacy of Frobeniusautomorphismen are the subject of the sentence tschebotarjowschen tightness. Ferdinand Georg Frobenius was the statement of the leak rate assumed in 1880, so the automorphisms are named after him.

Absolute and relative Frobenius for Schemes

Definition

Let be a prime number and a schema. The absolute Frobenius is defined as the identity on the topological space and potentiation on the structure sheaf. On an affine scheme of the absolute Frobenius is given by the Frobenius of the underlying ring, as can be seen in the global cuts. The fact that the prime ideals remain firm, translates into the equivalence.

Now let be a morphism of schemes over. The diagram

Commutes and induces the relative Frobeniusmorphismus

Of a morphism is over. Is the spectrum of a perfect ring, it is an isomorphism, then, but this is generally not a morphism isomorphism on.

Example

• With (above ), and the relative Frobenius is in coordinates given by:

• , Then, where it is meant that the coefficients are raised to the -th power. The relative Frobenius is induced by.

Properties

• Is whole, surjective and radiziell. For locally of finite presentation an isomorphism if is étale.
• When is flat, has the following local description: Be an open affine map. With the symmetric group and sit. The multiplication defines a ring homomorphism, and by gluing one obtains the schema.

Set of long-

A set of Serge Lang says: Be an algebraic affine or coherent group scheme over a finite field. Then the morphism is

Faithfully flat. Is algebra and commutative, so is an isogeny with core, long - isogeny. A corollary is that every - Torsor is trivial.

Examples:

• For one receives the Artin - Schreier - morphism.
• For obtaining the statement that any central simple algebra of rank is a matrix algebra over a finite field, for all that is put together the set of Wedderburn.

Frobenius and shift for commutative groups

Be a schema and a flat commutative group scheme. The above construction realized as a subscheme of the symmetric product (if it exists, otherwise you have with a smaller sub- scheme of work ), and by linking with the group multiplication one obtains a canonical morphism, the shift. The name comes from the fact that the shift in Witt vectors the picture

Is.

The following applies:

• Is a finite flat commutative group scheme, then swapped the Cartier duality and Frobenius shift:

A finite commutative group over a field if and only

• Of multiplicative type if is an isomorphism.
• étale if is an isomorphism.
• Infinitesimal, if for large.
• Unipotent if for large.

The characterizing group of properties and is the starting point of the Dieudonne theory.

Examples:

• For constant and groups.
• For diagonalizable groups.
• For the ordinary Frobeniushomomorphismus for rings. (Since the Frobeniusmorphismus is defined on the group structure without recourse, the inclusion is compatible with it. ) The shift is trivial.
• Is an abelian variety over a field of characteristic ( generally a abelian scheme), then the following sequence is exact if each case represents the core of the corresponding morphism:

Arithmetic and geometric Frobenius

Be a schema, on an algebraic closure of and. The Frobenius is called in this context arithmetic Frobenius, the inverse geometric Frobenius automorphism. Because is defined, and the relative Frobenius is. It is (even after the defining equation of the relative Frobenius )

Is a constant for cooking, inducing the identity of the cohomology, so that according to the above equation, the relative Frobenius with its coming out of the component geometry and the geometric Frobenius have the same effect.