Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general.
Definition.
Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism "F" is defined by
for all "r" in "R". It respects the multiplication of "R":
and is clearly 1 also. What is interesting, however, is that it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients
if . Therefore, the coefficients of all the terms except and are divisible by , the characteristic, and hence they vanish. Thus
This shows that "F" is a ring homomorphism.
If is a homomorphism of rings of characteristic , then 
If and are the Frobenius endomorphisms of and , 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 rings to itself.
If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: means , which by definition means that is nilpotent of order at most . In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective.
The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in . Then the image of does not contain . If it did, then there would be a rational function whose -th power would equal . But the degree of this -th power is , which is a multiple of . In particular, it can't be 1, which is the degree of . This is a contradiction; so is not in the image of .
A field 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.
Fixed points of the Frobenius endomorphism.
Consider the finite field . By Fermat's little theorem, every element of satisfies . Equivalently, it is a root of the polynomial . The elements of therefore determine roots of this equation, and because this equation has degree it has no more than roots over any extension. In particular, if is an algebraic extension of (such as the algebraic closure or another finite field), then is the fixed field of the Frobenius automorphism of .
Let be a ring of characteristic . If is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if is not a domain, then may have more than roots; for example, this happens if .
A similar property is enjoyed on the finite field formula_7 by the "n"th iterate of the Frobenius automorphism: Every element of formula_7 is a root of formula_9, so if is an algebraic extension of formula_7 and is the Frobenius automorphism of , then the fixed field of is formula_7. If "R" is a domain which is an formula_7-algebra, then the fixed points of the "n"th iterate of Frobenius are the elements of the image of formula_7.
Iterating the Frobenius map gives a sequence of elements in :
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 be the finite field of elements, where . The Frobenius automorphism of fixes the prime field , so it is an element of the Galois group . In fact, since formula_15 is cyclic with elements,
we know that the Galois group is cyclic and is a generator. The order of is because acts on an element by sending it to , and this is the identity on elements of . Every automorphism of is a power of , and the generators are the powers with coprime to .
Now consider the finite field as an extension of , where as above. If , then the Frobenius automorphism of does not fix the ground field , but its th iterate does. The Galois group is cyclic of order and is generated by . It is the subgroup of generated by . The generators of are the powers where is coprime to .
The Frobenius automorphism is not a generator of the absolute Galois group
because this Galois group is isomorphic to the profinite integers
which are not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of , 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 is a scheme of characteristic . Choose an open affine subset of . The ring is an -algebra, so it admits a Frobenius endomorphism. If is an open affine subset of , then by the naturality of Frobenius, the Frobenius morphism on , when restricted to , is the Frobenius morphism on . Consequently, the Frobenius morphism glues to give an endomorphism of . This endomorphism is called the absolute Frobenius morphism of , denoted . By definition, it is a homeomorphism of with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of -schemes to itself.
If is an -scheme and the Frobenius morphism of is the identity, then the absolute Frobenius morphism is a morphism of -schemes. In general, however, it is not. For example, consider the ring formula_18. Let and both equal with the structure map being the identity. The Frobenius morphism on sends to . It is not a morphism of formula_19-algebras. If it were, then multiplying by an element in formula_19 would commute with applying the Frobenius endomorphism. But this is not true because:
The former is the action of in the formula_19-algebra structure that begins with, and the latter is the action of formula_19 induced by Frobenius. Consequently, the Frobenius morphism on is not a morphism of formula_19-schemes.
The absolute Frobenius morphism is a purely inseparable morphism of degree . Its differential is zero. It preserves products, meaning that for any two schemes and , .
Restriction and extension of scalars by Frobenius.
Suppose that is the structure morphism for an -scheme . The base scheme has a Frobenius morphism "F""S". Composing with "F""S" results in an -scheme "X""F" called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an -morphism induces an -morphism .
For example, consider a ring "A" of characteristic and a finitely presented algebra over "A":
The action of "A" on "R" is given by:
where α is a multi-index. Let . Then is the affine scheme , but its structure morphism , and hence the action of "A" on "R", is different:
Because restriction of scalars by Frobenius is simply composition, many properties of are inherited by "X""F" under appropriate hypotheses on the Frobenius morphism. For example, if and "S""F" are both finite type, then so is "X""F".
The extension of scalars by Frobenius is defined to be:
The projection onto the factor makes an -scheme. If is not clear from the context, then is denoted by . Like restriction of scalars, extension of scalars is a functor: An -morphism determines an -morphism .
As before, consider a ring "A" and a finitely presented algebra "R" over "A", and again let . Then:
A global section of is of the form:
where "α" is a multi-index and every "a""iα" and "b""i" is an element of "A". The action of an element "c" of "A" on this section is:
Consequently, 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 has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does . 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 , there is a natural isomorphism:
Relative Frobenius.
Let be an -scheme with structure morphism . The relative Frobenius morphism of is the morphism:
defined by the universal property of the pullback (see the diagram above):
Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of -schemes.
Consider, for example, the "A"-algebra:
We have:
The relative Frobenius morphism is the homomorphism defined by:
Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of and , we have:
Relative Frobenius is a universal homeomorphism. If is an open immersion, then it is the identity. If is a closed immersion determined by an ideal sheaf "I" of , then is determined by the ideal sheaf and relative Frobenius is the augmentation map .
"X" is unramified over if and only if "F""X"/"S" is unramified and if and only if "F""X"/"S" is a monomorphism. "X" is étale over if and only if "F""X"/"S" is étale and if and only if "F""X"/"S" is an isomorphism.
Arithmetic Frobenius.
The arithmetic Frobenius morphism of an -scheme is a morphism:
defined by:
That is, it is the base change of "F""S" by 1"X".
Again, if:
then the arithmetic Frobenius is the homomorphism:
If we rewrite as:
then this homomorphism is:
Geometric Frobenius.
Assume that the absolute Frobenius morphism of is invertible with inverse formula_49. Let formula_50 denote the -scheme formula_51. Then there is an extension of scalars of by formula_49:
If:
then extending scalars by formula_49 gives:
If:
then we write:
and then there is an isomorphism:
The geometric Frobenius morphism of an -scheme is a morphism:
defined by:
It is the base change of formula_49 by .
Continuing our example of "A" and "R" above, geometric Frobenius is defined to be:
After rewriting "R" in terms of formula_64, geometric Frobenius is:
Arithmetic and geometric Frobenius as Galois actions.
Suppose that the Frobenius morphism of is an isomorphism. Then it generates a subgroup of the automorphism group of . If 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, and may be identified with . The arithmetic and geometric Frobenius morphisms are then endomorphisms of , and so they lead to an action of the Galois group of "k" on "X".
Consider the set of "K"-points . This set comes with a Galois action: Each such point "x" corresponds to a homomorphism 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
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 of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.
Suppose is an unramified extension of local fields, with ring of integers "OK" of such that the residue field, the integers of modulo their unique maximal ideal , is a finite field of order , where is a power of a prime. If is a prime of lying over , that is unramified means by definition that the integers of modulo , the residue field of , will be a finite field of order extending the residue field of where is the degree of . We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that
Frobenius for global fields.
In algebraic number theory, Frobenius elements are defined for extensions of global fields that are finite Galois extensions for prime ideals of that are unramified in . Since the extension is unramified the decomposition group of 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 as in the local case, by
where is the order of the residue field .
Lifts of the Frobenius are in correspondence with p-derivations.
Examples.
The polynomial
has discriminant
and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root of it to the field of -adic numbers gives an unramified extension of . We may find the image of under the Frobenius map by locating the root nearest to , which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the -adic integers . Modulo this polynomial is
This is algebraic over and is the correct global Frobenius image in terms of the embedding of into ; 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 -adic results will suffice.
If is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime in the base field . For an example, consider the extension of obtained by adjoining a root satisfying
to . This extension is cyclic of order five, with roots
for integer . It has roots which are Chebyshev polynomials of :
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 (which split). It is immediately apparent how the Frobenius map gives a result equal mod to the -th power of the root .