Frobenius homomorphism

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

Frobenius endomorphism of a ring

definition

Let it be a commutative unitary ring with the characteristic where is a prime number. As a Frobenius homomorphism, the mapping ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle p}$

${\displaystyle \phi _{p}\colon R\to R,\ \ x\mapsto x^{p}}$

designated. It is a ring homomorphism .

Is , then is too ${\displaystyle q=p^{e}}$

${\displaystyle \phi _{q}=\phi _{p^{e}}\colon R\to R,\ \ x\mapsto x^{q}}$

a ring homomorphism.

Proof of the homomorphism property

The figure is compatible with the multiplication in , because it is due to the power laws ${\displaystyle \phi _{p}}$ ${\displaystyle R}$

${\displaystyle \phi _{p}(x\cdot y)=(x\cdot y)^{p}=x^{p}\cdot y^{p}=\phi _{p}(x)\cdot \phi _{p}(y)}$

applies. Also applies Interestingly, the picture also with the addition in , that is acceptable, it is . With the help of the binomial theorem it follows ${\displaystyle \phi _{p}(1)=1^{p}=1.}$ ${\displaystyle R}$ ${\displaystyle \phi _{p}(x+y)=\phi _{p}(x)+\phi _{p}(y)}$

${\displaystyle (x+y)^{p}=x^{p}+\left(\sum _{k=1}^{p-1}{\binom {p}{k}}x^{p-k}y^{k}\right)+y^{p}}$

Since is a prime number , it does not divide for . Because the characteristic is therefore the numerator, but not the denominator of the binomial coefficients ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p!}$ ${\displaystyle m!}$ ${\displaystyle m<p}$ ${\displaystyle p}$

${\displaystyle {\binom {p}{k}}={\frac {p!}{k!(p-k)!}}}$

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

${\displaystyle (x+y)^{p}=x^{p}+y^{p}}$

and is compatible with the addition in . This equation is known in the English-speaking world as the Freshman's Dream (the beginner 's dream). ${\displaystyle R}$

use

The following is always a prime number and a power of . All rings or bodies that occur have characteristics . ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p}$

• After the little Fermat's theorem is on the residue class ring the identity . More generally: if there is a finite body , then the identity is. ${\displaystyle \phi _{p}}$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle \phi _{q}}$
• If there is a body then is . ${\displaystyle K}$ ${\displaystyle \{x\in K:\phi _{p}(x)=x\}=\mathbb {F} _{p}}$
• If an extension is a finite field, then is an automorphism of which leaves element-wise fixed. The Galois group is cyclic and is generated by. ${\displaystyle \mathbb {F} _{q^{n}}/\mathbb {F} _{q}}$ ${\displaystyle \phi _{q}}$ ${\displaystyle \mathbb {F} _{q^{n}}}$ ${\displaystyle \mathbb {F} _{q}}$ ${\displaystyle {\text{Gal}}(\mathbb {F} _{q^{n}}/\mathbb {F} _{q})}$ ${\displaystyle \phi _{q}}$
• Is a ring, then is injective if and only if it contains no non-trivial nilpotent elements . (The gist of is .) ${\displaystyle A}$ ${\displaystyle \phi _{p}\colon A\to A}$ ${\displaystyle A}$ ${\displaystyle \phi _{p}}$ ${\displaystyle \{a\in A:a^{p}=0\}}$
• If a ring is and is bijective, then the ring is called perfect (or perfect). In a perfect ring, each element has an uniquely defined root. Perfect bodies are characterized by the fact that they have no inseparable extensions. ${\displaystyle A}$ ${\displaystyle \phi _{p}\colon A\to A}$ ${\displaystyle p}$
• The perfect conclusion of a ring can be represented as an inductive limit : ${\displaystyle A}$

${\displaystyle A^{p^{-\infty }}=\varinjlim \left(A{\stackrel {\phi _{p}}{\longrightarrow }}A{\stackrel {\phi _{p}}{\longrightarrow }}A{\stackrel {\phi _{p}}{\longrightarrow }}\dots \right)}$

• The additivity of mapping is also used in the Artin-Schreier theory . ${\displaystyle x\mapsto x^{p}}$

Frobenius automorphisms of local and global bodies

The following assumptions serve to describe the case of a finite Galois expansion of algebraic number fields as well as local fields . Let be a Dedekind ring , its quotient field , a finite Galois expansion , the whole closure of in . Then there is a dedekind ring. Let further be a maximal ideal in with finite remainder class field , furthermore and . The body enlargement is Galois. Be the Galois group of . It operates transitively on the overlying prime ideals of . Let be the decomposition group , i.e. H. the stabilizer of . The induced homomorphism ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle L}$ ${\displaystyle B}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle B}$ ${\displaystyle \lambda =B/{\mathfrak {P}}}$ ${\displaystyle {\mathfrak {p}}={\mathfrak {P}}\cap A}$ ${\displaystyle \kappa =A/{\mathfrak {p}}}$ ${\displaystyle \lambda /\kappa }$ ${\displaystyle G}$ ${\displaystyle L/K}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle B}$ ${\displaystyle G_{\mathfrak {P}}}$ ${\displaystyle {\mathfrak {P}}}$

${\displaystyle r:G_{\mathfrak {P}}\to {\text{Gal}}(\lambda /\kappa )}$

is surjective. Its core is the inertia group .

It is now unbranched , i.e. H. . Then the homomorphism is an isomorphism. The Frobenius automorphism (also Frobenius element) is the archetype of the Frobenius automorphism below . It is clearly characterized by the following property: ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {p}}B_{\mathfrak {P}}={\mathfrak {P}}}$ ${\displaystyle r}$ ${\displaystyle {\text{Frob}}_{\mathfrak {P}}\in {\text{Gal}}(L/K)}$ ${\displaystyle \phi _{|\kappa |}\in {\text{Gal}}(\lambda /\kappa )}$ ${\displaystyle r}$

${\displaystyle {\text{Frob}}_{\mathfrak {P}}b\equiv b^{|\kappa |}\mod {\mathfrak {P}}}$

Because the prime ideals operate transitively, the Frobenius automorphisms are conjugated to them , so that their conjugation class is uniquely determined by. If the extension is Abelian , one obtains a clear Frobenius automorphism . ${\displaystyle G}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle L/K}$ ${\displaystyle {\text{Frob}}_{\mathfrak {p}}\in {\text{Gal}}(L/K)}$

Frobenius automorphisms are of central importance for class field theory : In the ideal theoretical formulation, the reciprocity mapping is induced by the assignment . Conjugation classes of Frobenius automorphisms are the subject of Chebotaryov's density theorem . Ferdinand Georg Frobenius had already suspected the statement of the density theorem in 1880, which is why the automorphisms are named after him. ${\displaystyle {\mathfrak {p}}\mapsto {\text{Frob}}_{\mathfrak {p}}}$

Absolute and relative Frobenius for schemes

definition

Let be a prime number and a scheme over . The absolute Frobenius is defined as identity on the topological space and -potenzierung on the structural grain . On an affine schema , the absolute Frobenius is given by the Frobenius of the underlying ring, as can be seen from the global sections. The fact that the prime ideals remain fixed translates into equivalence . ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \phi _{X}\colon X\to X}$ ${\displaystyle p}$ ${\displaystyle {\text{Spec }}A}$ ${\displaystyle a\in {\mathfrak {p}}\iff a^{p}\in {\mathfrak {p}}}$

Now be a morphism of schemes over . The diagram ${\displaystyle X\to S}$ ${\displaystyle \mathbb {F} _{p}}$

${\displaystyle {\begin{matrix}X&{\stackrel {\phi _{X}}{\longrightarrow }}&X\\\downarrow &&\downarrow \\S&{\stackrel {\phi _{S}}{\longrightarrow }}&S\end{matrix}}}$

commutes and induces the relative Frobenius morphism

${\displaystyle F_{X/S}\colon X\to X^{(p/S)}=S\times _{\phi _{S},S}X}$

which is a morphism about . If the spectrum is a perfect ring , then there is an isomorphism, well , but this isomorphism is generally not a morphism about . ${\displaystyle S}$ ${\displaystyle S={\text{Spec }}A}$ ${\displaystyle A}$ ${\displaystyle \phi _{S}}$ ${\displaystyle X^{(p/S)}\cong X}$ ${\displaystyle S}$

example

• With is (over ), and the relative Frobenius is given in coordinates by: ${\displaystyle X=S[T_{1},\dots ,T_{n}]}$ ${\displaystyle X^{(p)}\cong S[T_{1},\dots ,T_{n}]}$ ${\displaystyle S}$

${\displaystyle T_{i}\mapsto T_{i}^{p}}$

• Is , then is , where is intended to mean that the coefficients are raised to the power of -th. The relative Frobenius is induced by. ${\displaystyle B=A[T_{1},\dots ,T_{n}]/(f_{1},\dots ,f_{m})}$ ${\displaystyle ({\text{Spec }}B)^{(p/{\text{Spec }}A)}=A[T_{1}^{p},\dots ,T_{n}^{p}]/({\tilde {f}}_{1},\dots ,{\tilde {f}}_{m})}$ ${\displaystyle {\tilde {f}}}$ ${\displaystyle p}$ ${\displaystyle ({\text{Spec }}B)^{(p/{\text{Spec }}A)}\to B}$ ${\displaystyle T_{i}\mapsto T_{i}^{p}}$

properties

• ${\displaystyle F_{X/S}}$is whole, surjective and radical. For locally of finite presentation there is an isomorphism if and only if étale is. ${\displaystyle X/S}$ ${\displaystyle F_{X/S}}$ ${\displaystyle X/S}$
• If is flat, has the following local description: Let is an open affine map of . With the symmetrical group and place . The multiplication defines a ring homomorphism , and gluing together gives the scheme . ${\displaystyle X/S}$ ${\displaystyle X^{(p/S)}}$ ${\displaystyle {\text{Spec }}A}$ ${\displaystyle X}$ ${\displaystyle S_{p}}$ ${\displaystyle \textstyle N=\sum _{\sigma \in S_{p}}\sigma }$ ${\displaystyle A^{(p)}=(A^{\otimes p})^{S_{p}}/N\cdot A^{\otimes p}}$ ${\displaystyle A^{(p)}\to A}$ ${\displaystyle {\text{Spec }}A^{(p)}}$ ${\displaystyle X^{(p)}}$

Lang's theorem

A sentence by Serge Lang says: Let be an algebraic or affine connected group scheme over a finite field . Then there is morphism ${\displaystyle G}$ ${\displaystyle \mathbb {F} _{q}}$

${\displaystyle L:x\mapsto x^{-1}\cdot F_{q}(x)}$

extremely flat. Is algebraic and commutative, so it is an isogeny with a nucleus , the Lang isogeny. One corollary is that everyone - torsor is trivial. ${\displaystyle G}$ ${\displaystyle L}$ ${\displaystyle G(\mathbb {F} _{q})}$ ${\displaystyle G}$

Examples:

• For one obtains the Artin-Schreier morphism . ${\displaystyle G=\mathbb {G} _{a}}$
• For one obtains the statement that every central simple algebra of rank over a finite field is a matrix algebra , thus for all taken together Wedderburn's theorem . ${\displaystyle G={\text{GL}}_{n}}$ ${\displaystyle n}$ ${\displaystyle n}$

Frobenius and displacement for commutative groups

Be a scheme and a flat commutative group scheme . The above construction is realized as a sub-scheme of the symmetrical product (if this exists, otherwise one has to work with a smaller sub- scheme of), and by concatenation with the group multiplication one obtains a canonical morphism , the displacement. The name comes from the fact that the shift in Wittvectors is the mapping ${\displaystyle S}$ ${\displaystyle G/S}$ ${\displaystyle G^{(p/S)}}$ ${\displaystyle G^{p}/S_{p}}$ ${\displaystyle G^{p}}$ ${\displaystyle V_{G/S}\colon G^{(p/S)}\to G}$

${\displaystyle (x_{0},x_{1},x_{2},\dots )\mapsto (0,x_{0},x_{1},\dots )}$

is.

The following applies:

• ${\displaystyle V_{G/S}\circ F_{G/S}=p,\ \ F_{G/S}\circ V_{G/S}=p}$

(Multiplication with in the group or ). ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle G^{(p)}}$

• ${\displaystyle {\text{Lie}}(G/S)={\text{Lie}}(\ker(F_{G/S})/S)}$
• If there is a finite flat commutative group scheme, then the Cartier duality exchanges Frobenius and displacement: ${\displaystyle G/S}$

${\displaystyle F_{D(G)/S}=D(V_{G/S}),\ \ V_{D(G)/S}=D(F_{G/S})}$

A finite commutative group over a body is then ${\displaystyle G}$

• of the multiplicative type when is an isomorphism. ${\displaystyle V}$
• étale when is an isomorphism. ${\displaystyle F}$
• infinitesimal if for large. ${\displaystyle F^{n}=0\colon G\to G^{(p^{n})}=((G^{(p)})\dots )^{(p)}}$ ${\displaystyle n}$
• unipotent if for big. ${\displaystyle V^{n}=0\colon G^{(p^{n})}\to G}$ ${\displaystyle n}$

The characterization of groups by properties of and is the starting point of the Dieudonné theory . ${\displaystyle F}$ ${\displaystyle V}$

Examples:

• For constant groups is and . ${\displaystyle F={\text{id}}}$ ${\displaystyle V=p}$
• For diagonalizable groups is and . ${\displaystyle F=p}$ ${\displaystyle V={\text{id}}}$
• For is the ordinary Frobenius homomorphism for rings . (Since the Frobeniusmorphismus is defined without recourse to the group structure, the inclusion is compatible with it.) The shift is trivial: . ${\displaystyle G=\mathbb {G} _{a}}$ ${\displaystyle F}$ ${\displaystyle \phi _{p}\colon A\to A}$ ${\displaystyle A=\mathbb {G} _{a}(A)}$ ${\displaystyle \mathbb {G} _{m}(A)\subseteq \mathbb {G} _{a}(A)}$ ${\displaystyle V=0}$
• If an Abelian variety is over a field of the characteristic (more generally an Abelian scheme), then the following sequence is exact if it stands for the core of the corresponding morphism : ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle {}_{F}X}$ ${\displaystyle F\colon X\to Y}$

${\displaystyle 0\to {}_{F^{n}}X\to {}_{p^{n}}X{\stackrel {F^{n}}{\longrightarrow }}{}_{V^{n}}X^{(p^{n})}\to 0}$

Arithmetic and Geometric Frobenius

Let be a schema over , further an algebraic closure of and . The Frobenius is called in this context arithmetic Frobenius, the inverse automorphism geometric Frobenius. Because over is defined, is , and the relative Frobenius is . The following applies (also according to the defining equation of the relative Frobenius) ${\displaystyle X}$ ${\displaystyle k=\mathbb {F} _{q}}$ ${\displaystyle {\bar {k}}}$ ${\displaystyle k}$ ${\displaystyle {\overline {X}}=X\times _{{\text{Spec }}k}{\text{Spec }}{\bar {k}}}$ ${\displaystyle \phi _{q}\in {\text{Gal}}({\bar {k}}/k)}$ ${\displaystyle \phi _{q}^{-1}}$ ${\displaystyle {\overline {X}}}$ ${\displaystyle k}$ ${\displaystyle {\overline {X}}^{(q/{\bar {k}})}\cong {\overline {X}}}$ ${\displaystyle F_{{\overline {X}}/{\bar {k}}}=\phi _{q,X}\times {\text{id}}_{\bar {k}}}$

${\displaystyle \phi _{q,{\overline {X}}}=({\text{id}}_{X}\times \phi _{q,{\bar {k}}})\circ (\phi _{q,X}\times {\text{id}}_{\bar {k}})}$

If a constant sheaf is on , the identity induces on the cohomology of , so that according to the above equation the relative Frobenius with its component coming from geometry and the geometric Frobenius have the same effect. ${\displaystyle G}$ ${\displaystyle {\overline {X}}_{\text{et}}}$ ${\displaystyle \phi _{q,{\overline {X}}}}$ ${\displaystyle G}$ ${\displaystyle \phi _{q,X}\times {\text{id}}_{\bar {k}}}$ ${\displaystyle \phi _{q,X}}$ ${\displaystyle {\text{id}}_{X}\times \phi _{q,{\bar {k}}}^{-1}}$

literature

• Serge Lang: Algebra (=  Graduate Texts in Mathematics . Volume 211 ). 3. Edition. Springer, New York 2002, ISBN 0-387-95385-X .
• Michel Demazure, Pierre Gabriel: Groupes algébriques. Tome 1 . North-Holland, Amsterdam 1970, ISBN 978-0-7204-2034-0 .
• Pierre Gabriel: Expose VII _A . Étude infinitesimale des schémas en groupes . In: Michel Demazure, Alexander Grothendieck (ed.): Séminaire de Géométrie Algébrique du Bois-Marie 1962–1964 (SGA 3): Schémas en groupes. Tome 1: Propriétés générales des schémas en groupes . Springer, Berlin 1970, ISBN 978-3-540-05180-0 .
• Christian Houzel: Exposé XV. Morphisme de Frobenius et des fonctions rationalité L . In: Luc Illusie (ed.): Séminaire de Géometrie Algébrique du Bois-Marie 1965-66 (SGA 5): Cohomologie l-adique et Fonctions L (=  Lecture Notes in Mathematics ). tape 589 . Springer, Berlin 1977, ISBN 3-540-08248-4 .

Footnotes

1. ↑ V §1 Definition 2 in: Nicolas Bourbaki: Elements of Mathematics. Algebra II. Chapters 4-7 . Springer, Berlin 2003, ISBN 978-3-540-00706-7 .
2. ^ Lang, VII §2
3. ^ Peter Stevenhagen, Hendrik Lenstra: Chebotarëv and his density theorem . In: Mathematical Intelligencer . tape 18 , no. 2 , 1996, p. 26-37 . The original work is: Georg Ferdinand Frobenius: About relationships between the prime ideals of an algebraic field and the substitutions of its group . In: Session reports of the Royal Prussian Academy of Sciences in Berlin . 1896, p. 689-703 .
4. ^ Houzel, §1 Proposition 2
5. ^ Gabriel, 4.2
6. ↑ Demazure-Gabriel, III §5, 7.2. The original work is: Serge Lang: Algebraic Groups Over Finite Fields . In: Amer. J. Math. Band 78 , no. 3 , 1956, pp. 555-563 .
7. ↑ Demazure-Gabriel, II §7
8. ^ Proposition 2.3 in: Tadao Oda: The first de Rham cohomology group and Dieudonné modules . In: Annales scientifiques de l'École Normale Supérieure, Sér. 4 . tape 2 , no. 1 , 1969, p. 63-135 ( online ).
9. ↑ Houzel, §2 Proposition 2