Witt vector

Witt vectors are an introduced by the mathematician Ernst Witt generalization of the construction of the ( whole ) of p- adic numbers to arbitrary perfect residue field. In addition to these typical Witt vectors there is the big Witt vectors, which give information about the typical Witt vectors for any reconstructed.

• 2.1 Definition
• 2.2 Alternative definition of power series
• 2.3 Frobenius and shift
• 2.4 Relationship to the p- typical Witt vectors Artin - Hasse exponential
• 2.5 λ -rings
• 2.6 Cartier theory

• 4.1 Textbooks and review articles
• 4.2 Advanced Topics

P- typical Witt vectors

Let be a fixed prime number. For a ring ( commutative, with unit element ) form the Witt vectors of a dependent evident ring. It is especially interesting for rings of characteristic design but makes it necessary to allow other rings.

Motivation

Be an integer. As an approximation of an alternative construction of the first -adic number is only by using the addition and multiplication in the body of a residue class ring of isomorphous ring, denoted by, can be constructed.

The first, naive approach to this would be to use the illustration that the residue class maps integers from in the residue class of in. The bijection

Corresponds to the representation of integers in the place value system to the base. The addition is then transmitted from the case:

Wherein the transfer is. This construction can not be good on other body as generalized, but also because of the definition of the unfavorable view of algebraic representative system makes use.

The correct approach is based on the following statement from elementary number theory: For integers:

(see congruence ( number theory ) ). This means that the actual and a representative of, the residue class of depends to only, not on the choice of starting. We write suggestive of this element of. (This figure is essentially the Teichmüller system of representatives for the - adic numbers. ) General also depends on the residue class of not saying we washers.

Because each case is bijective, we get by adding up a bijective mapping:

Be the crowd together with that of addition and multiplication that make it an isomorphism.

Now let special and so. If two vectors are added and, therefore, we obtain modulo the equation, ie. In order to

The polynomial

Has coefficients divisible, so is equal to a polynomial. In order to

For a total of

The associativity of addition is translated into an equation

It is easy to assume that this equation is already considered accordingly. This means that for an arbitrary commutative ring by setting

Can define the structure of an abelian group. The same applies to

With, so that becomes a commutative ring with unit element.

Definition

Denote the set of non-negative integers. Furthermore, is a fixed prime number.

There are clearly certain polynomials for each such that the following holds for every commutative ring with unit element: is a ring with addition:

And multiplication

And for each, the mapping

A ring homomorphism. means of the ring - typical Witt vectors with entries. If only the speech of typical Witt vectors is only written.

For with the corresponding cut-off addition and multiplication also be a commutative ring with unit element of the ring of Witt vectors of length typical.

The ring element

Is called the -th ghost component or sub-component of. With the Witt polynomials

Can be calculated recursively and:

Examples:

The negation in the ring is given by universal polynomials. For is:

For on the other hand with

The figure is multiplicative and is called Teichmüller representative system (after Oswald Teichmüller ).

Sketch of proof

The recursive description provides. In order, on the other hand prove one hand, the integers, the ring features the classic proof approach is more general:

Lemma. Is a polynomial (for example ), then there are unique polynomials with integer

For everyone. Corresponding versions of this statement apply to take or even.

Rational Uniqueness is clear, the Ganzzahligkeitsbeweis based on the properties and as well as the above-mentioned implication

The ring properties of follow from the uniqueness assertion of the lemma: Both and are given by polynomials which are solutions of the following equation:

So these polynomials are equal.

Another proof approach used to identify the ring of big Witt vectors with the ring, see below.

Simple properties

• Can be identified with, and with the projection. All projections are surjective ring homomorphisms, and

• And
• If is invertible, then the picture on the ghost components is a ring isomorphism.
• Other examples ( under both isomorphisms corresponds to the vector ):

W (k) for k perfect body

Be a perfect field of characteristic. Then is a complete discrete valuation ring of mixed characteristic ( ie ), whose maximal ideal is generated by. This property characterizes up to isomorphism.

Witt vectors play an important role in the structure theory of complete local rings ( according to IS Cohen ):

• Set of Teichmüller Witt: Is a complete noetherian local ring with residue field, then there exists a unique homomorphism such that the concatenation with the projection is equal to the projection. There is exactly one multiplicative section of the projection, called Teichmüller representative system, and the picture is:

• Than - algebra is isomorphic to a quotient of with.
• Is not a zero divisor in, then there is with elements such that the induced homomorphism is injective and is a module finitely generated.
• In the special case of the more accurate means: Is a complete discrete valuation ring of characteristic 0 with residue field, then is a finite extension of the degree, if the normalized score of is so true.

For non- perfect body Cohen- rings take on the role of.

Frobenius and shift

In characteristic p

Be a ring of characteristic. The shift is the picture

She is a homomorphism of additive groups. By cutting to get induced homomorphisms

The Frobeniushomomorphismus (based on the Frobeniushomomorphismus of fields of characteristic ) is the mapping

It is a ring homomorphism which restricts to ring homomorphisms. Be the multiplication by on. Then

Thus

In particular

It should also

Frobenius and shift are special cases of a more general construction, see Frobeniushomomorphismus # shift.

Be the quotient field of. Then the ( arithmetic ) Frobenius for the field extension.

Dieudonné ring

Be a perfect field of characteristic. If we write and for the module in which the module structure is given by, then one obtains module homomorphisms

In analogy to Frobenius and shift for algebraic groups in characteristic. More generally, if a module with two module homomorphisms and, one can summarize this structure as a module for the Dieudonné ring ( after Jean Dieudonné ), the non-commutative ring generated by two symbols and, with the relations

The classical Dieudonné theory is an equivalence of categories between commutative unipotent algebraic groups and certain moduli. See also below.

Generally

For the definition of the rings, any Frobeniushomomorphismus has to be modified: it is characterized by the equation. In particular, the 0 -th component. The Frobeniushomomorphismus is a ring homomorphism in the general case. It is

By cutting one obtains ring homomorphisms

(that is no longer as in the case of the target pattern). In general, still

And

Frobeniuslifts and Komonadenstruktur

Be a - torsion-free ring. A Frobeniuslift is a ring homomorphism with. For a Frobeniuslift exists after Dieudonné -Cartier is a uniquely determined continuation, applies for all. You met. Since even has the Frobeniuslift, initially obtained for torsion - rings and universal formulas for arbitrary rings a natural transformation which is characterized by. It is also called Artin - Hasse exponential function, see also below, and defines a Komonade.

The restriction on torsion - rings can be eliminated by the fact that one goes to - derivations: For a ring A - Derivation is a figure for the picture

Is a ring homomorphism. Specifically, this means that the following equations satisfied:

A - derivation defined on by a Frobeniuslift. Is torsion, we get reversed from a Frobeniuslift a Derivation

A ring with one - Derivation is called δ - ring.

The situation is so far analogous to ordinary derivations, as these can be characterized by the fact that a ring homomorphism is.

The coalgebras for the above defined Komonade can be identified with the δ -rings. Specifically rechtsadjungiert for forgetful of the category of the δ -rings in the category of the rings. There is also a dual description based on the " Plethorie " representing as Endofunktor category of the rings.

More properties in characteristic p

Be a ring with.

• If an integral domain, then also, and it is.
• The units of are precisely the elements.
• When a body is, then, a local ring with maximum ideal. In addition, if and noetherian if it is perfect.
• If is surjective, then and thus.
• Is perfect, that is, bijective, then can be a Witt vector with the Teichmüller mapping write as - adically convergent series:

• Is an integral domain, and are all primes in invertible (eg if a body is ), then one can describe the unit groups and (formal power series and Laurent series ) as well as, see below.

Other applications

• Artin - Schreier -Witt theory: If a field of characteristic, abelian extensions of exponent can be classified using the Witt vectors.
• Is a schematic diagram over a field of characteristic, then there is not always a flat schema. The existence of a lift after playing a role in the proof of the degeneration of the Hodge -de Rham spectral sequence of Pierre Deligne and Luc Illusie.
• Is smooth, lifts exist locally. Equips you with the local lift even a PD- structure which causes an analogue of the Poincaré lemma applies, we obtain the crystalline cohomology. The crystalline cohomology groups are -modules. You Tensoriert with quotient field, we obtain a Weil cohomology, the l -adic cohomology for supplementary.
• If a schema, the topological space with the sheaf is a scheme again. The de Rham -Witt complex is a suitable quotient of. For smooth crystalline cohomology is isomorphic to Hyperkohomologie of.
• There are approaches to apply Witt vectors on the analysis of the encryption method NTRUEncrypt.

Witt vectors as algebraic group

Be a perfect field of characteristic. The Witt vectors of length form a commutative algebraic group as a variety isomorphic to affine space. is a unipotent group: This follows from the filtration with subquotients or the Artin - Hasse - embedding.

In characteristic 0 every commutative unipotent group is isomorphic to. In positive characteristic, the theory is much more complex: there are non-trivial extensions, and unless there is still the possible composition factors and (the core of the Frobeniusmorphismus on explicitly ).

Every commutative unipotent group over is isogenic to a product of Witt vector groups. The fundamental theorem of classical Dieudonné theory goes that the functor

Defined by an equivalence between the category of commutative unipotent algebraic groups and the category of finitely generated -modules on which acts nilpotent. With the help of the Cartier duality or with Witt covectors can construct an analogous equivalence for finite groups as well as for p- divisible groups.

For an abelian variety, there is a canonical isomorphism of -modules. The core of multiplication by, and the algebraic De Rham cohomology is. The - divisible group of the Dieudonné module is isomorphic to the crystalline cohomology.

Witt covectors

How Witt vectors are a generalization of the p-adic numbers, so are Witt covectors a generalization of the auditor group. The functor enables a uniform presentation of Dieudonné theory for finite commutative groups and divisible groups over a perfect body.

For a ring, the direct limit is of

This becomes an Ind - group scheme. In older literature, the symbol will be used. ie the group of unipotent Witt covectors.

The construction of the topological group of all Witt covectors is more complicated elements are identified in can with consequences that are from an index zero. With the same universal formulas, it is for consequences which have from a fixed index values ​​in a fixed nilpotent ideal, declare an addition. Equip these groups with the product topology of discrete factors and laws. The unipotent covectors form a dense subset of.

Be a perfect ring of characteristic and a - algebra. The figure

Makes it a module (different from the above-defined module structure), and with the Frobenius and the shift to a module. The displacement is linear, and is obtained in a module structure.

Branched Witt vectors

Be a complete discrete valuation ring of characteristic 0 with Uniformisierender whose residue field is a finite field with elements. Then there is a functorial algebra structure for algebras, so that

For each a homomorphism of algebras is. There are Frobenius and shift operators, by

Are characterized. For a finite expansion of the residual body is a straight extension of the degree. Branched Witt vectors assume the role of the ordinary Witt vectors in the transmission of the Cartier theory of formal moduli.

Large Witt vectors

Definition

Denote the set of positive integers.

There are clearly certain polynomials such that the following holds for every commutative ring with unit element: is a ring with addition

And multiplication

And for each, the mapping

A ring homomorphism. Also, the additive inverse is given by universal polynomials. means of the ring of Witt vectors with large or universal entries.

Is a subset, so that is also for each divisor of in, then with the corresponding cut-off addition and multiplication of also a commutative ring with unit element. For one obtains the ring of big Witt vectors of length, for a prime number is obtained up to Umindizierung the ring of Witt vectors typical, see below.

The ring element

Is called the -th ghost component or sub-component of. With the Witt polynomials

Can be calculated recursively and:

The figure is multiplicative and is called Teichmüller representative system.

Can be represented as functor -valued by a polynomial ring in countably many indeterminates. In practice, one actually used the ring of symmetric polynomials, and transfer of structures.

Alternative definition with power series

Be the multiplicative group of formal power series with constant term 1 The figure

Is an isomorphism of groups. For has

As coefficients of the components of spirits.

Under the isomorphism of the product of two Witt vectors is mapped to:

Where in each case. Write for the multiplication in corresponding link, so that an isomorphism of rings. As a special case of the multiplication formula results

Frobenius and shift

There are operators and for each natural number. Their effect on the spirits of components is:

In is

Here, a formal primitive root of unity, and the norm. In particular

For the multiplication is on with, so

If an algebra (in particular ), then there is an operator for each:

It goes for:

In the last formula stands for the ith component of.

Relationship to the p- typical Witt vectors Artin - Hasse exponential

Let be a prime number. The figure is a surjective ring homomorphism. The typical Wittpolynome are under this Umindizierung equal to the big Wittpolynomen, the same reasoning applies also to the spirits components.

The subset is not a subring of. In certain cases, however, one can embed in.

The Artin - Hasse exponential

Can be regarded as an element of ( ie have the coefficients not divisible by the denominator, see localization; is the Möbius function).

If an algebra, that is, are all primes invertible, then for a Witt vector element

Well defined. is an idempotent in, and induces a ring isomorphism. Denote the corresponding subset of with. Then:

The ring decomposes as a direct product of the for. For any rings when the ratio of designated, which can not be obtained by projection onto the components with index divisible by.

Frobenius and shift restrict to operators on and tune on there with the declared operators or match.

For a field of characteristic is the one-unit group of, and we obtain the isomorphism

For each algebra

Cutting off at reduced to a factor of, with the smallest integer having. One thus gets an isomorphism of algebraic groups (over )

The Artin - Hasse exponential is also related to the Komonadenstruktur together: For a perfect field of characteristic is the concatenation of

With the projection of the same.

λ -rings

There is a canonical ring homomorphism satisfying. If the abelian group is torsion, is uniquely determined by this condition and is characterized for other rings that the equation is valid for a surjection with a torsion ring. Along with becoming a Komonade. Applying the coalgebras to this Komonade on, you get the so -called λ -rings.

The first component corresponds to spirits in the first coefficient:

A pre- λ - ring is a ring together with a group homomorphism with. This condition is the compatibility with the Koeins the Komonade. Denoting the coefficients of with, so

Then a pre- λ structure equivalent to specify images for which satisfy the following equations:

A λ - homomorphism is a ring homomorphism, ie the following diagram commutes:

The ring has as stated above for each ring a canonical pre- λ structure. A λ - ring is a pre- λ - ring for the λ is a homomorphism. The above chart is just for compatibility with the comultiplication of Komonade. Translated into which are the additional conditions of the form:

The ( universal ) polynomials describe the multiplication and possess as the polynomials a description with the help of elementary symmetric polynomials.

The Koassoziativität the Komonade way that even a λ - ring. The functor is the forgetful rechtsadjungiert of the category of λ -rings in the category of rings.

If a λ - ring, then the ring homomorphism

The - te on Adams operation. It is true. For a prime number, so it is, that is, is a Frobeniuslift. Is an arbitrary ring, then the operation on the Adams λ Ring Frobenius.

Cartier- theory

The Cartier theory ( according to Pierre Cartier ) is an equivalence of categories between a suitable category of commutative formal groups over a ring and a sub-category of modules over the Cartier ring.

Be the category of commutative algebras without identity, which consist only of nilpotent elements. For the purposes of the theory of commutative formal groups with functors can be identified. Is a formal group law, the corresponding functor of an algebra assigns the lot to the group structure. The formal group is the formal affine line. The functors can be continued in a natural way on the category of algebras, the filtered projective Limites of algebras are.

The formal group of Witt vectors is the functor that an algebra the subset of Witt vectors in maps that only a finite number of 0 different components. The corresponding sub- group consists of the elements with respect to a polynomial. The ring is called with designated and Cartier ring. The operators restrict to endomorphisms of a and thus define elements. The designations are swapped to and from, so the above relations hold back because of the multiplication reversed order. The figure is an injective ring homomorphism.

Be a formal group. The following groups are naturally isomorphic:

• The group of morphisms (not group homomorphisms, ie natural transformations only as set-valued functors ). The group structure is induced by the group structure.
• The group of homomorphisms

Its elements are called curves in the group denoted by. From the last description, a canonical links module structure is on.

The power series group can be identified with. Witt the polynomials corresponding to the homomorphism, which is induced by the logarithmic derivative of.

In the operation of induced by the operation of of. For a formal look - th root of unity again and form into the sum of the curves obtained by for. For a curve is determined by the image of the coordinate. One identifies with the corresponding element in the effects of defined with the above agreement agree (without interchanging and ).

Both and carry natural topologies. The fundamental theorem of Cartier theory says that an equivalence between a category of formal groups over and a category of topological moduli induced. The inverse functor assigns a module to a suitable constructed tensor.

Let be a prime number and a - algebra, that is, every prime is invertible. Then an idempotent is in, sit. A module for the subgroup of the elements is all. Such elements are called typical.

For a formal group (where the formal group of the - typical Witt vectors, analogous to ) the group of typical curves was. Then induces an equivalence between the category of formal groups over as above, and one category of topological -modules. The inverse functor is a tensor product as before.

For a perfect field of characteristic of the Dieudonné ring can be identified with a dense ring. Under suitable conditions the Dieudonné module is dual to the typical curves module.

Generalizations

• Colette Schoeller has extended parts of the typical theory, namely the construction of the Cohen- ring and the classification of unipotent groups on non- perfect body.
• Andreas Dress and Christian Siebeneicher have specified the construction of a ring of a profinite group and a ring such that is isomorphic to the completed Burnside ring of is. For arises for results.