Frobenius endomorphism

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{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic Template:Mvar, an important class which includes finite fields. The endomorphism maps every element to its Template:Mvar-th power. In certain contexts it is an automorphism, but this is not true in general.

Definition

Let Template:Mvar be a commutative ring with prime characteristic Template:Mvar (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by

for all r in R. Clearly this respects the multiplication of R:

and F(1) is clearly 1 also. What is interesting, however, is that it also respects the addition of Template:Mvar. The expression (r + s)p can be expanded using the binomial theorem. Because Template:Mvar is prime, it divides p! but not any q! for q < p; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients

if 1 ≤ k ≤ p − 1. Therefore the coefficients of all the terms except rp and sp are divisible by Template:Mvar, the characteristic, and hence they vanish.[1] Thus

This shows that F is a ring homomorphism.

If φ : RS is a homomorphism of rings of characteristic Template:Mvar, then:

If FR and FS are the Frobenius endomorphisms of Template:Mvar and Template:Mvar, then this can be rewritten as:

This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic Template:Mvar rings to itself.

If the ring Template:Mvar is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: F(r) = 0 means rp = 0, which by definition means that Template:Mvar is nilpotent of order at most Template:Mvar. In fact, this is an if and only if, because if Template:Mvar is any nilpotent, then one of its powers will be nilpotent of order at most Template:Mvar. In particular, if Template:Mvar is a field then the Frobenius endomorphism is injective.

The Frobenius morphism is not necessarily surjective, even when Template:Mvar is a field. For example let K = Fp(t) be the finite field of Template:Mvar elements together with a single transcendental element; equivalently, Template:Mvar is the field of rational functions with coefficients in Fp. Then the image of Template:Mvar does not contain Template:Mvar. If it did, then there would be a rational function q(t)/r(t) whose Template:Mvar-th power q(t)p/r(t)p would equal Template:Mvar. But the degree of this Template:Mvar-th power is p deg(q) − p deg(r), which is a multiple of Template:Mvar. In particular, it can't be 1, which is the degree of Template:Mvar. This is a contradiction, so Template:Mvar is not in the image of Template:Mvar.

A field Template:Mvar is called perfect if either it is of characteristic zero or if it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect.

Fixed points of the Frobenius endomorphism

Consider the finite field Fp. By Fermat's little theorem, every element Template:Mvar of Fp satisfies xp = x. Equivalently, it is a root of the polynomial XpX. The elements of Fp therefore determine Template:Mvar roots of this equation, and because this equation has degree Template:Mvar it has no more than Template:Mvar roots over any extension. In particular, if Template:Mvar is an algebraic extension of Fp (such as the algebraic closure or another finite field), then Fp is the fixed field of the Frobenius automorphism of Template:Mvar.

Let Template:Mvar be a ring of characteristic p > 0. If Template:Mvar is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if Template:Mvar is not a domain, then XpX may have more than Template:Mvar roots; for example, this happens if R = Fp × Fp.

A similar property is enjoyed on the finite field by the eth iterate of the Frobenius automorphism: Every element of is a root of , so if Template:Mvar is an algebraic extension of and Template:Mvar is the Frobenius automorphism of Template:Mvar, then the fixed field of Fe is . If R is a domain which is an -algebra, then the fixed points of the eth iterate of Frobenius are the elements of the image of .

Iterating the Frobenius map gives a sequence of elements in Template:Mvar:

This sequence of iterates is used in defining the Frobenius closure and the tight closure of an ideal.

As a generator of Galois groups

The Galois group of an extension of finite fields is generated by an iterate of the Frobenius automorphism. First, consider the case where the ground field is the prime field. Let Fq be the finite field of Template:Mvar elements, where q = pe. The Frobenius automorphism Template:Mvar of Fq fixes the prime field Fp, so it is an element of the Galois group Gal(Fq/Fp). In fact, this Galois group is cyclic and Template:Mvar is a generator. The order of Template:Mvar is Template:Mvar because F e acts on an element Template:Mvar by sending it to xq, and this is the identity on elements of Fq. Every automorphism of Fq is a power of Template:Mvar, and the generators are the powers F i with Template:Mvar coprime to Template:Mvar.

Now consider the finite field Fq f as an extension of Fq. The Frobenius automorphism Template:Mvar of Fq f does not fix the ground field Fq, but its Template:Mvar-th iterate F e does. The Galois group Gal(Fq f /Fq) is cyclic of order Template:Mvar and is generated by F e. It is the subgroup of Gal(Fq f /Fp) generated by F e. The generators of Gal(Fq f /Fq) are the powers F ei where Template:Mvar is coprime to Template:Mvar.

The Frobenius automorphism is not a generator of the absolute Galois group

because this Galois group is

which is not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of Fq, it is a generator of every finite quotient of the absolute Galois group. Consequently it is a topological generator in the usual Krull topology on the absolute Galois group.

Frobenius for schemes

There are several different ways to define the Frobenius morphism for a scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations.

The absolute Frobenius morphism

Suppose that Template:Mvar is a scheme of characteristic p > 0. Choose an open affine subset U = Spec A of Template:Mvar. The ring Template:Mvar is an Fp-algebra, so it admits a Frobenius endomorphism. If Template:Mvar is an open affine subset of Template:Mvar, then by the naturality of Frobenius, the Frobenius morphism on Template:Mvar, when restricted to Template:Mvar, is the Frobenius morphism on Template:Mvar. Consequently the Frobenius morphism glues to give an endomorphism of Template:Mvar. This endomorphism is called the absolute Frobenius morphism of Template:Mvar. By definition, it is a homeomorphism of Template:Mvar with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of Fp-schemes to itself.

If Template:Mvar is an Template:Mvar-scheme and the Frobenius morphism of Template:Mvar is the identity, then the absolute Frobenius morphism is a morphism of Template:Mvar-schemes. In general, however, it is not. For example, consider the ring . Let Template:Mvar and Template:Mvar both equal Spec A with the structure map XS being the identity. The Frobenius morphism on Template:Mvar sends Template:Mvar to ap. It is not a morphism of -algebras. If it were, then multiplying by an element Template:Mvar in would commute with applying the Frobenius endomorphism. But this is not true because:

The former is the action of Template:Mvar in the -algebra structure that Template:Mvar begins with, and the latter is the action of induced by Frobenius. Consequently, the Frobenius morphism on Spec A is not a morphism of -schemes.

The absolute Frobenius morphism is a purely inseparable morphism of degree Template:Mvar. Its differential is zero. It preserves products, meaning that for any two schemes Template:Mvar and Template:Mvar, FX×Y = FX × FY.

Restriction and extension of scalars by Frobenius

Suppose that φ : XS is the structure morphism for an Template:Mvar-scheme Template:Mvar. The base scheme Template:Mvar has a Frobenius morphism FS. Composing Template:Mvar with FS results in an Template:Mvar-scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an Template:Mvar-morphism XY induces an Template:Mvar-morphism XFYF.

For example, consider a ring A of characteristic p > 0 and a finitely presented algebra over A:

The action of A on R is given by:

where α is a multi-index. Let X = Spec R. Then XF is the affine scheme Spec R, but its structure morphism Spec R → Spec A, and hence the action of A on R, is different:

Because restriction of scalars by Frobenius is simply composition, many properties of Template:Mvar are inherited by XF under appropriate hypotheses on the Frobenius morphism. For example, if Template:Mvar and SF are both finite type, then so is XF.

The extension of scalars by Frobenius is defined to be:

The projection onto the Template:Mvar factor makes X(p) an Template:Mvar-scheme. If Template:Mvar is not clear from the context, then X(p) is denoted by X(p/S). Like restriction of scalars, extension of scalars is a functor: An Template:Mvar-morphism XY determines an Template:Mvar-morphism X(p)Y(p).

As before, consider a ring A and a finitely presented algebra R over A, and again let X = Spec R. Then:

A global section of X(p) is of the form:

where α is a multi-index and every aiα and bi is an element of A. The action of an element c of A on this section is:

Consequently, X(p) is isomorphic to:

where, if:

then:

A similar description holds for arbitrary A-algebras R.

Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if Template:Mvar has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does X(p). Furthermore, being a base change means that extension of scalars preserves properties such as being of finite type, finite presentation, separated, affine, and so on.

Extension of scalars is well-behaved with respect to base change: Given a morphism S′ → S, there is a natural isomorphism:

Relative Frobenius

The relative Frobenius morphism of an Template:Mvar-scheme X is the morphism:

defined by:

Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of Template:Mvar-schemes.

Consider, for example, the A-algebra:

We have:

The relative Frobenius morphism is the homomorphism R(p)R defined by:

Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of X(p/S) ×S S and (X ×S S′)(p/S′), we have:

Relative Frobenius is a universal homeomorphism. If XS is an open immersion, then it is the identity. If XS is a closed immersion determined by an ideal sheaf I of OS, then X(p) is determined by the ideal sheaf Ip and relative Frobenius is the augmentation map OS/IpOS/I.

X is unramified over Template:Mvar if and only if FX/S is unramified and if and only if FX/S is a monomorphism. X is étale over Template:Mvar if and only if FX/S is étale and if and only if FX/S is an isomorphism.

Arithmetic Frobenius

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The arithmetic Frobenius morphism of an Template:Mvar-scheme Template:Mvar is a morphism:

defined by:

That is, it is the base change of FS by 1X.

Again, if:

then the arithmetic Frobenius is the homomorphism:

If we rewrite R(p) as:

then this homomorphism is:

Geometric Frobenius

Assume that the absolute Frobenius morphism of Template:Mvar is invertible with inverse . Let denote the Template:Mvar-scheme . Then there is an extension of scalars of Template:Mvar by :

If:

then extending scalars by gives:

If:

then we write:

and then there is an isomorphism:

The geometric Frobenius morphism of an Template:Mvar-scheme Template:Mvar is a morphism:

defined by:

It is the base change of by 1X.

Continuing our example of A and R above, geometric Frobenius is defined to be:

After rewriting R(1/p) in terms of , geometric Frobenius is:

Arithmetic and geometric Frobenius as Galois actions

Suppose that the Frobenius morphism of Template:Mvar is an isomorphism. Then it generates a subgroup of the automorphism of group of Template:Mvar. If S = Spec k is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, X(p) and X(1/p) may be identified with Template:Mvar. The arithmetic and geometric Frobenius morphisms are then endomorphisms of Template:Mvar, and so they lead to an action of the Galois group of k on X.

Consider the set of K-points X(K). This set comes with a Galois action: Each such point x corresponds to a homomorphism OXk(x) ≅ K from the structure sheaf to the residue field at x, and the action of Frobenius on x is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism

is the same as the composite morphism:

by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of X.

Frobenius for local fields

Given an unramified finite extension L/K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.[2]

Suppose L/K is an unramified extension of local fields, with ring of integers OK of Template:Mvar such that the residue field, the integers of Template:Mvar modulo their unique maximal ideal Template:Mvar, is a finite field of order Template:Mvar. If Φ is a prime of Template:Mvar lying over Template:Mvar, that L/K is unramified means by definition that the integers of Template:Mvar modulo Φ, the residue field of Template:Mvar, will be a finite field of order qf extending the residue field of Template:Mvar where Template:Mvar is the degree of L/K. We may define the Frobenius map for elements of the ring of integers OL of Template:Mvar as an automorphism sΦ of Template:Mvar such that

Frobenius for global fields

In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of Template:Mvar that are unramified in L/K. Since the extension is unramified the decomposition group of Φ is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of Template:Mvar as in the local case, by

where Template:Mvar is the order of the residue field OK mod Φ.

Lifts of the Frobenius are in correspondence with p-derivations.

Examples

The polynomial

x5x − 1

has discriminant

19 × 151,

and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root Template:Mvar of it to the field of 3-adic numbers Q3 gives an unramified extension Q3(ρ) of Q3. We may find the image of Template:Mvar under the Frobenius map by locating the root nearest to ρ3, which we may do by Newton's method. We obtain an element of the ring of integers Z3[ρ] in this way; this is a polynomial of degree four in Template:Mvar with coefficients in the 3-adic integers Z3. Modulo 38 this polynomial is

.

This is algebraic over Q and is the correct global Frobenius image in terms of the embedding of Q into Q3; moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if Template:Mvar-adic results will suffice.

If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime Template:Mvar in the base field Template:Mvar. For an example, consider the extension Q(β) of Q obtained by adjoining a root Template:Mvar satisfying

to Q. This extension is cyclic of order five, with roots

for integer Template:Mvar. It has roots which are Chebyshev polynomials of Template:Mvar:

β2 − 2, β3 − 3β, β5 − 5β3 + 5β

give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form 22n + 1 (which split). It is immediately apparent how the Frobenius map gives a result equal mod Template:Mvar to the Template:Mvar-th power of the root Template:Mvar.

See also

References

  1. This is known as the Freshman's dream.
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