In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
and
field theory, the Frobenius endomorphism (after
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
) is a special
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
with prime
characteristic , an important class which includes
finite fields
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtra ...
. The endomorphism maps every element to its -th power. In certain contexts it is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
, but this is not true in general.
Definition
Let be a commutative ring with prime characteristic (an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
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 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
. Because is prime, it divides but not any for ; it therefore will divide the
numerator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, but not the
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, of the explicit formula of the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s
:
if . Therefore, the coefficients of all the terms except and are divisible by , and hence they vanish.
[This is known as the ]Freshman's dream
The freshman's dream is a name sometimes given to the erroneous equation (x+y)^n=x^n+y^n, where n is a real number (usually a positive integer greater than 1) and x,y are nonzero real numbers. Beginning students commonly make this error in computi ...
. 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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
from the identity
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
on the category of characteristic rings to itself.
If the ring is a ring with no
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
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
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, 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
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
, 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
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
. 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
by the ''n''th iterate of the Frobenius automorphism: Every element of
is a root of
, so if is an algebraic extension of
and is the Frobenius automorphism of , then the fixed field of is
. If ''R'' is a domain which is an
-algebra, then the fixed points of the ''n''th iterate of Frobenius are the elements of the image of
.
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 In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by .
Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p i ...
of an ideal.
As a generator of Galois groups
The
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
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
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
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...
:
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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
. 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
. Let and both equal with the structure map being the identity. The Frobenius morphism on sends to . It is not a morphism of
-algebras. If it were, then multiplying by an element in
would commute with applying the Frobenius endomorphism. But this is not true because:
:
The former is the action of in the
-algebra structure that begins with, and the latter is the action of
induced by Frobenius. Consequently, the Frobenius morphism on is not a morphism of
-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
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
(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
. Let
denote the -scheme
. Then there is an extension of scalars of by
:
:
If:
:
then extending scalars by
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
by .
Continuing our example of ''A'' and ''R'' above, geometric Frobenius is defined to be:
:
After rewriting ''R'' in terms of
, 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
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
finite extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...
of
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
s.
Suppose is an unramified extension of local fields, with
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
''O
K'' 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
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, Frobenius elements are defined for extensions of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s that are finite
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
s for
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s 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
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
:,
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
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
. 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
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshe ...
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 .
See also
*
Perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k'' is ...
*
Frobenioid In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of Fro ...
*
*
Universal homeomorphism In algebraic geometry, a universal homeomorphism is a morphism of schemes f: X \to Y such that, for each morphism Y' \to Y, the base change X \times_Y Y' \to Y' is a homeomorphism of topological spaces.
A morphism of schemes is a universal homeomor ...
References
*
*{{Springer, id=f/f041770, title=Frobenius endomorphism
Finite fields
Algebraic number theory
Galois theory