Template:Short description Template:Refimprove 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 that 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.
DefinitionEdit
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
- <math>F(r) = r^p</math>
for all r in R. It respects the multiplication of R:
- <math>F(rs) = (rs)^p = r^ps^p = F(r)F(s),</math>
and Template:Math is 1 as well. Moreover, it also respects the addition of Template:Mvar. The expression Template:Math can be expanded using the binomial theorem. Because Template:Mvar is prime, it divides Template:Math but not any Template:Math for Template:Math; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients
- <math>\frac{p!}{k! (p-k)!},</math>
if Template:Math. Therefore, the coefficients of all the terms except Template:Math and Template:Math are divisible by Template:Mvar, and hence they vanish.<ref>This is known as the freshman's dream.</ref> Thus
- <math>F(r + s) = (r + s)^p = r^p + s^p = F(r) + F(s).</math>
This shows that F is a ring homomorphism.
If Template:Math is a homomorphism of rings of characteristic Template:Mvar, then
- <math>\varphi(x^p) = \varphi(x)^p.</math>
If Template:Math and Template:Math are the Frobenius endomorphisms of Template:Mvar and Template:Mvar, then this can be rewritten as:
- <math>\varphi \circ F_R = F_S \circ \varphi.</math>
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: Template:Math means Template:Math, which by definition means that Template:Mvar is nilpotent of order at most Template:Mvar. In fact, this is necessary and sufficient, 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 Template:Math 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 Template:Math. Then the image of Template:Mvar does not contain Template:Mvar. If it did, then there would be a rational function Template:Math whose Template:Mvar-th power Template:Math would equal Template:Mvar. But the degree of this Template:Mvar-th power (the difference between the degrees of its numerator and denominator) is Template:Math, 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 it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect.
Fixed points of the Frobenius endomorphismEdit
Consider the finite field Template:Math. By Fermat's little theorem, every element Template:Mvar of Template:Math satisfies Template:Math. Equivalently, it is a root of the polynomial Template:Math. The elements of Template:Math 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 Template:Math (such as the algebraic closure or another finite field), then Template:Math is the fixed field of the Frobenius automorphism of Template:Mvar.
Let Template:Mvar be a ring of characteristic Template:Math. 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 Template:Math may have more than Template:Mvar roots; for example, this happens if Template:Math.
A similar property is enjoyed on the finite field <math>\mathbf{F}_{p^n}</math> by the nth iterate of the Frobenius automorphism: Every element of <math>\mathbf{F}_{p^n}</math> is a root of <math>X^{p^n} - X</math>, so if Template:Mvar is an algebraic extension of <math>\mathbf{F}_{p^n}</math> and Template:Mvar is the Frobenius automorphism of Template:Mvar, then the fixed field of Template:Math is <math>\mathbf{F}_{p^n}</math>. If R is a domain that is an <math>\mathbf{F}_{p^n}</math>-algebra, then the fixed points of the nth iterate of Frobenius are the elements of the image of <math>\mathbf{F}_{p^n}</math>.
Iterating the Frobenius map gives a sequence of elements in Template:Mvar:
- <math>x, x^p, x^{p^2}, x^{p^3}, \ldots.</math>
This sequence of iterates is used in defining the Frobenius closure and the tight closure of an ideal.
As a generator of Galois groupsEdit
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 Template:Math. Let Template:Math be the finite field of Template:Mvar elements, where Template:Math. The Frobenius automorphism Template:Mvar of Template:Math fixes the prime field Template:Math, so it is an element of the Galois group Template:Math. In fact, since <math>\mathbf{F}_q^{\times}</math> is [[Finite_field#Multiplicative_structure|cyclic with Template:Nowrap elements]], we know that the Galois group is cyclic and Template:Mvar is a generator. The order of Template:Mvar is Template:Mvar because Template:Math acts on an element Template:Mvar by sending it to Template:Math, and <math>x^{p^j} = x</math> can only have <math>p^j</math> many roots, since we are in a field. Every automorphism of Template:Math is a power of Template:Mvar, and the generators are the powers Template:Math with Template:Mvar coprime to Template:Mvar.
Now consider the finite field Template:Math as an extension of Template:Math, where Template:Math as above. If Template:Math, then the Frobenius automorphism Template:Mvar of Template:Math does not fix the ground field Template:Math, but its Template:Mvarth iterate Template:Math does. The Galois group Template:Math is cyclic of order Template:Mvar and is generated by Template:Math. It is the subgroup of Template:Math generated by Template:Math. The generators of Template:Math are the powers Template:Math where Template:Mvar is coprime to Template:Mvar.
The Frobenius automorphism is not a generator of the absolute Galois group
- <math>\operatorname{Gal} \left (\overline{\mathbf{F}_q}/\mathbf{F}_q \right ),</math>
because this Galois group is isomorphic to the profinite integers
- <math>\widehat{\mathbf{Z}} = \varprojlim_n \mathbf{Z}/n\mathbf{Z},</math>
which are not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of Template:Math, 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 schemesEdit
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 morphismEdit
Suppose that Template:Mvar is a scheme of characteristic Template:Math. Choose an open affine subset Template:Math of Template:Mvar. The ring Template:Mvar is an Template:Math-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, denoted Template:Math. 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 Template:Math-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 <math>A = \mathbf{F}_{p^2}</math>. Let Template:Mvar and Template:Mvar both equal Template:Math with the structure map Template:Math being the identity. The Frobenius morphism on Template:Mvar sends Template:Mvar to Template:Math. It is not a morphism of <math>\mathbf{F}_{p^2}</math>-algebras. If it were, then multiplying by an element Template:Mvar in <math>\mathbf{F}_{p^2}</math> would commute with applying the Frobenius endomorphism. But this is not true because:
- <math>b \cdot a = ba \neq F(b) \cdot a = b^p a.</math>
The former is the action of Template:Mvar in the <math>\mathbf{F}_{p^2}</math>-algebra structure that Template:Mvar begins with, and the latter is the action of <math>\mathbf{F}_{p^2}</math> induced by Frobenius. Consequently, the Frobenius morphism on Template:Math is not a morphism of <math>\mathbf{F}_{p^2}</math>-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, Template:Math.
Restriction and extension of scalars by FrobeniusEdit
Suppose that Template:Math 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 Template:Math induces an Template:Mvar-morphism Template:Math.
For example, consider a ring A of characteristic Template:Math and a finitely presented algebra over A:
- <math>R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m).</math>
The action of A on R is given by:
- <math>c \cdot \sum a_\alpha X^\alpha = \sum c a_\alpha X^\alpha,</math>
where α is a multi-index. Let Template:Math. Then Template:Math is the affine scheme Template:Math, but its structure morphism Template:Math, and hence the action of A on R, is different:
- <math>c \cdot \sum a_\alpha X^\alpha = \sum F(c) a_\alpha X^\alpha = \sum c^p a_\alpha X^\alpha.</math>
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:
- <math>X^{(p)} = X \times_S S_F.</math>
The projection onto the Template:Mvar factor makes Template:Math an Template:Mvar-scheme. If Template:Mvar is not clear from the context, then Template:Math is denoted by Template:Math. Like restriction of scalars, extension of scalars is a functor: An Template:Mvar-morphism Template:Math determines an Template:Mvar-morphism Template:Math.
As before, consider a ring A and a finitely presented algebra R over A, and again let Template:Math. Then:
- <math>X^{(p)} = \operatorname{Spec} R \otimes_A A_F.</math>
A global section of Template:Math is of the form:
- <math>\sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i = \sum_i \sum_\alpha X^\alpha \otimes a_{i\alpha}^p b_i,</math>
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:
- <math>c \cdot \sum_i \left (\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i = \sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i c.</math>
Consequently, Template:Math is isomorphic to:
- <math>\operatorname{Spec} A[X_1, \ldots, X_n] / \left (f_1^{(p)}, \ldots, f_m^{(p)} \right ),</math>
where, if:
- <math>f_j = \sum_\beta f_{j\beta} X^\beta,</math>
then:
- <math>f_j^{(p)} = \sum_\beta f_{j\beta}^p X^\beta.</math>
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 Template:Math. 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 Template:Math, there is a natural isomorphism:
- <math>X^{(p/S)} \times_S S' \cong (X \times_S S')^{(p/S')}.</math>
Relative FrobeniusEdit
Let Template:Math be an Template:Math-scheme with structure morphism Template:Math. The relative Frobenius morphism of Template:Math is the morphism:
- <math>F_{X/S} : X \to X^{(p)}</math>
defined by the universal property of the pullback Template:Math (see the diagram above):
- <math>F_{X/S} = (F_X, \varphi).</math>
Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of Template:Mvar-schemes.
Consider, for example, the A-algebra:
- <math>R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m).</math>
We have:
- <math>R^{(p)} = A[X_1, \ldots, X_n] / (f_1^{(p)}, \ldots, f_m^{(p)}).</math>
The relative Frobenius morphism is the homomorphism Template:Math defined by:
- <math>\sum_i \sum_\alpha X^\alpha \otimes a_{i\alpha} \mapsto \sum_i \sum_\alpha a_{i\alpha}X^{p\alpha}.</math>
Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of Template:Math and Template:Math, we have:
- <math>F_{X / S} \times 1_{S'} = F_{X \times_S S' / S'}.</math>
Relative Frobenius is a universal homeomorphism. If Template:Math is an open immersion, then it is the identity. If Template:Math is a closed immersion determined by an ideal sheaf I of Template:Math, then Template:Math is determined by the ideal sheaf Template:Math and relative Frobenius is the augmentation map Template:Math.
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 FrobeniusEdit
The arithmetic Frobenius morphism of an Template:Mvar-scheme Template:Mvar is a morphism:
- <math>F^a_{X/S} : X^{(p)} \to X \times_S S \cong X</math>
defined by:
- <math>F^a_{X/S} = 1_X \times F_S.</math>
That is, it is the base change of FS by 1X.
Again, if:
- <math>R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m),</math>
- <math>R^{(p)} = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m) \otimes_A A_F,</math>
then the arithmetic Frobenius is the homomorphism:
- <math>\sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i \mapsto \sum_i \sum_\alpha a_{i\alpha} b_i^p X^\alpha.</math>
If we rewrite Template:Math as:
- <math>R^{(p)} = A[X_1, \ldots, X_n] / \left (f_1^{(p)}, \ldots, f_m^{(p)} \right ),</math>
then this homomorphism is:
- <math>\sum a_\alpha X^\alpha \mapsto \sum a_\alpha^p X^\alpha.</math>
Geometric FrobeniusEdit
Assume that the absolute Frobenius morphism of Template:Mvar is invertible with inverse <math>F_S^{-1}</math>. Let <math>S_{F^{-1}}</math> denote the Template:Mvar-scheme <math>F_S^{-1} : S \to S</math>. Then there is an extension of scalars of Template:Mvar by <math>F_S^{-1}</math>:
- <math>X^{(1/p)} = X \times_S S_{F^{-1}}.</math>
If:
- <math>R = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m),</math>
then extending scalars by <math>F_S^{-1}</math> gives:
- <math>R^{(1/p)} = A[X_1, \ldots, X_n] / (f_1, \ldots, f_m) \otimes_A A_{F^{-1}}.</math>
If:
- <math>f_j = \sum_\beta f_{j\beta} X^\beta,</math>
then we write:
- <math>f_j^{(1/p)} = \sum_\beta f_{j\beta}^{1/p} X^\beta,</math>
and then there is an isomorphism:
- <math>R^{(1/p)} \cong A[X_1, \ldots, X_n] / (f_1^{(1/p)}, \ldots, f_m^{(1/p)}).</math>
The geometric Frobenius morphism of an Template:Mvar-scheme Template:Mvar is a morphism:
- <math>F^g_{X/S} : X^{(1/p)} \to X \times_S S \cong X</math>
defined by:
- <math>F^g_{X/S} = 1_X \times F_S^{-1}.</math>
It is the base change of <math>F_S^{-1}</math> by Template:Math.
Continuing our example of A and R above, geometric Frobenius is defined to be:
- <math>\sum_i \left(\sum_\alpha a_{i\alpha} X^\alpha\right) \otimes b_i \mapsto \sum_i \sum_\alpha a_{i\alpha} b_i^{1/p} X^\alpha.</math>
After rewriting RTemplate:I sup in terms of <math>\{f_j^{(1/p)}\}</math>, geometric Frobenius is:
- <math>\sum a_\alpha X^\alpha \mapsto \sum a_\alpha^{1/p} X^\alpha.</math>
Arithmetic and geometric Frobenius as Galois actionsEdit
Suppose that the Frobenius morphism of Template:Mvar is an isomorphism. Then it generates a subgroup of the automorphism group of Template:Mvar. If Template:Math 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, Template:Math and Template:Math 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 Template:Math. This set comes with a Galois action: Each such point x corresponds to a homomorphism Template:Math 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
- <math>\mathcal{O}_X \to k(x) \xrightarrow{\overset{}F} k(x)</math>
is the same as the composite morphism:
- <math>\mathcal{O}_X \xrightarrow{\overset{}F^a_{X/S}} \mathcal{O}_X \to k(x)</math>
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 fieldsEdit
Given an unramified finite extension Template:Math of local fields, there is a concept of Frobenius endomorphism that induces the Frobenius endomorphism in the corresponding extension of residue fields.<ref name=FT144>Template:Cite book</ref>
Suppose Template:Math 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, where Template:Mvar is a power of a prime. If Template:Math is a prime of Template:Mvar lying over Template:Mvar, that Template:Math is unramified means by definition that the integers of Template:Mvar modulo Template:Math, the residue field of Template:Mvar, will be a finite field of order Template:Math extending the residue field of Template:Mvar where Template:Mvar is the degree of Template:Math. We may define the Frobenius map for elements of the ring of integers Template:Math of Template:Mvar as an automorphism Template:Math of Template:Mvar such that
- <math>s_\Phi(x) \equiv x^q \mod \Phi.</math>
Frobenius for global fieldsEdit
In algebraic number theory, Frobenius elements are defined for extensions Template:Math of global fields that are finite Galois extensions for prime ideals Template:Math of Template:Mvar that are unramified in Template:Math. Since the extension is unramified the decomposition group of Template:Math 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
- <math>s_\Phi(x) \equiv x^q \mod \Phi,</math>
where Template:Mvar is the order of the residue field Template:Math.
Lifts of the Frobenius are in correspondence with p-derivations.
ExamplesEdit
The polynomial
has discriminant
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 Template:Math-adic numbers Template:Math gives an unramified extension Template:Math of Template:Math. We may find the image of Template:Mvar under the Frobenius map by locating the root nearest to Template:Math, which we may do by Newton's method. We obtain an element of the ring of integers Template:Math in this way; this is a polynomial of degree four in Template:Mvar with coefficients in the Template:Math-adic integers Template:Math. Modulo Template:Math this polynomial is
- <math>\rho^3 + 3(460+183\rho-354\rho^2-979\rho^3-575\rho^4)</math>.
This is algebraic over Template:Math and is the correct global Frobenius image in terms of the embedding of Template:Math into Template:Math; 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 Template:Math 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 Template:Math of Template:Math obtained by adjoining a root Template:Mvar satisfying
- <math>\beta^5+\beta^4-4\beta^3-3\beta^2+3\beta+1=0</math>
to Template:Math. This extension is cyclic of order five, with roots
- <math>2 \cos \tfrac{2 \pi n}{11}</math>
for integer Template:Mvar. It has roots that are Chebyshev polynomials of Template:Mvar:
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 Template:Math (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.