Frobenius endomorphism

In a ring with prime characteristic p, the map raising elements to the pth power

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) 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 p-th power. In certain contexts it is an automorphism, but this is not true in general.

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

and F(1) is 1 as well. However, it also respects the addition of R. The expression (r + s)p can be expanded using the binomial theorem. Because p 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 ≤ kp − 1. Therefore, the coefficients of all the terms except rp and sp are divisible by p, and hence they vanish.[1] Thus

If FR and FS are the Frobenius endomorphisms of R and S, 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 p rings to itself.

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

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

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

Consider the finite field Fp. By Fermat's little theorem, every element x of Fp satisfies xp = x. Equivalently, it is a root of the polynomial XpX. The elements of Fp therefore determine p roots of this equation, and because this equation has degree p it has no more than p roots over any extension. In particular, if K 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 K.

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

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

Now consider the finite field Fqf as an extension of Fq, where q = pn as above. If n > 1, then the Frobenius automorphism F of Fqf does not fix the ground field Fq, but its nth iterate Fn does. The Galois group Gal(Fqf /Fq) is cyclic of order f and is generated by Fn. It is the subgroup of Gal(Fqf /Fp) generated by Fn. The generators of Gal(Fqf /Fq) are the powers Fni where i is coprime to f.

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

which are 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.

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.

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

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

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

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

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 X are inherited by XF under appropriate hypotheses on the Frobenius morphism. For example, if X and SF are both finite type, then so is XF.

The projection onto the S factor makes X(p) an S-scheme. If S 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 S-morphism XY determines an S-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:

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

Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if X 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:

Let X be an S-scheme with structure morphism φ. The relative Frobenius morphism of X is the morphism:

defined by the universal property of the pullback X(p) (see the diagram above):

Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of S-schemes.

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

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

Suppose that the Frobenius morphism of S is an isomorphism. Then it generates a subgroup of the automorphism group of S. 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 X. The arithmetic and geometric Frobenius morphisms are then endomorphisms of X, 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 from the structure sheaf to K, which factors via k(x), 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

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.

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 K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of order q, where q is a power of a prime. If Φ is a prime of L lying over φ, that L/K is unramified means by definition that the integers of L modulo Φ, the residue field of L, will be a finite field of order qf extending the residue field of K where f is the degree of L/K. We may define the Frobenius map for elements of the ring of integers OL of L as an automorphism sΦ of L such that

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 L 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 L as in the local case, by

and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root ρ of it to the field of 3-adic numbers Q3 gives an unramified extension Q3(ρ) of Q3. We may find the image of ρ 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 ρ 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 p-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 φ in the base field K. For an example, consider the extension Q(β) of Q obtained by adjoining a root β satisfying

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 p to the p-th power of the root β.