Witt vector

In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.

History 

In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummer theory. Let k be a field containing a primitive nth root of unity. Kummer theory classifies degree n cyclic field extensions K of k. Such fields are in bijection with order n cyclic groups , where  corresponds to .

But suppose that k has characteristic p. The problem of studying degree p extensions of k, or more generally degree pn extensions, may appear superficially similar to Kummer theory. However, in this situation, k cannot contain a primitive pth root of unity. If x is a pth root of unity in k, then it satisfies . Because raising to the pth power is the Frobenius homomorphism, this equation may be rewritten as , and therefore . Consequently Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.

The case where the characteristic divides the degree is now called Artin–Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two. This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree p extensions of a field k of characteristic p were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form . By repeating their construction, they described degree p2 extensions. Abraham Adrian Albert used this idea to describe degree pn extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.

Schmid generalized further to non-commutative cyclic algebras of degree pn. In the process of doing so, certain polynomials related to addition of p-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree pn field extensions and cyclic algebras. Specifically, he introduced a ring now called Wn(k), the ring of n-truncated p-typical Witt vectors. This ring has k as a quotient, and it comes with an operator F which is called the Frobenius operator because it reduces to the Frobenius operator on k. Witt observes that the degree pn analog of Artin–Schreier polynomials is

where . To complete the analogy with Kummer theory, define  to be the operator . Then the degree pn extensions of k are in bijective correspondence with cyclic subgroups  of order pn, where  corresponds to the field .

Motivation

Any -adic integer (an element of , not to be confused with ) can be written as a power series , where the 's are usually taken from the integer interval . It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients  is only one of many choices, and Hensel himself (the creator of p-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number  together with the  roots of unity; that is, the solutions of  in , so that . This choice extends naturally to ring extensions of  in which the residue field is enlarged to  with , some power of . Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the  solutions in the field to . Call the field , with  an appropriate primitive  root of unity (over ). The representatives are then  and  for .
Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field  of order  by taking residues modulo  in , and elements of  are taken to their representatives by the Teichmüller character . This operation identifies the set of integers in  with infinite sequences of elements of .

Taking those representatives the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: ): given two infinite sequences of elements of  describe their sum and product as -adic integers explicitly. This problem was solved by Witt using Witt vectors.

Detailed motivational sketch

We derive the ring of -adic integers  from the finite field  using a construction which naturally generalizes to the Witt vector construction.

The ring  of -adic integers can be understood as the projective limit of  Specifically, it consists of the sequences  with  such that  for  That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections .

The elements of  can be expanded as (formal) power series in 

 

where  are usually taken from the integer interval  Of course, this power series usually will not converge in  using the standard metric on the reals, but it will converge in  with the p-adic metric. We will sketch a method of defining ring operations for such power series.

Letting  be denoted by , one might consider the following definition for addition:

and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set 

There is a better coefficient subset of which does yield closed formulas, the Teichmuller representatives: zero together with the roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives )  as roots of  through Hensel lifting, the p-adic version of Newton's method. For example, in  to calculate the representative of , one starts by finding the unique solution of  in  with ; one gets . Repeat this in , with the conditions  and  gives  and so on; the resulting Teichmüller representative is the sequence  The existence of a lift in each step is guaranteed by the greatest common divisor  in every 

This algorithm shows that for every , there is exactly one Teichmuller representative with , which we denote  Indeed, this defines the Teichmüller character satisfying  if we denote  Note that  is not additive, as the sum need not be a representative. Despite this, if  in  then  in  

Because of this one-to-one correspondence given by , one can expand every -adic integer as a power series in  with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as  Then, if one has some arbitrary p-adic integer of the form  one takes the difference  leaving a value divisible by . Hence, . The process is then repeated, subtracting  and proceed likewise. This yields a sequence of congruences

So that 

and  implies: 

for 

Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than . It is clear that 

since 

 

for all  as  so the difference tends to 0 with respect to the p-adic metric. The resulting coefficients will typically differ from the 's modulo , except the first one.

The Teichmuller coefficients have the key additional property that  which is missing for the numbers in . This can be used to describe addition, as follows. Since the Teichmüller character is not additive,  is not true in . But it holds in  as the first congruence implies. In particular, 

and thus

Since the binomial coefficient  is divisible by , this gives

This completely determines  by the lift. Moreover, the congruence modulo  indicates that the calculation can actually be done in  satisfying the basic aim of defining a simple additive structure.

For  this step is already very cumbersome. Write

Just as for  a single th power is not enough: one must take

However,  is not in general divisible by  but it is divisible when  in which case  combined with similar monomials in  will make a multiple of .

At this step, it becomes clear that one is actually working with addition of the form

This motivates the definition of Witt vectors.

Construction of Witt rings
Fix a prime number p. A Witt vector over a commutative ring R is a sequence:  of elements of R. Define the Witt polynomials  by 

and in general

The  are called the ghost components of the Witt vector , and are usually denoted by The ghost components can be thought of as an alternative coordinate system for the R-module of sequences. 

The ring of Witt vectors is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring such that:

the sum and product are given by polynomials with integral coefficients that do not depend on R, and

projection to each ghost component is a ring homomorphism from the Witt vectors over R, to R.

In other words, 

 and  are given by polynomials with integral coefficients that do not depend on R, and

 and .

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example, 

.

These are to be understood as shortcuts for the actual formulas. If for example the ring R has characteristic p, the division by p in the first formula above, the one by  that would appear in the next component and so forth, do not make sense. However, if the p-power of the sum is developed, the terms  are cancelled with the previous ones and the remaining ones are simplified by p, no division by p remains and the formula makes sense. The same consideration applies to the ensuing components.

Examples
The Witt ring of any commutative ring R in which p is invertible is just isomorphic to  (the product of a countable number of copies of R). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to , and if p is invertible this homomorphism is an isomorphism.

The Witt ring of the finite field of order p is the ring of p-adic integers written in terms of the Teichmuller representatives, as demonstrated above.

The Witt ring of a finite field of order pn is the unramified extension of degree n of the ring of p-adic integers.

Universal Witt vectors
The Witt polynomials for different primes p are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime p). 
Define the universal Witt polynomials Wn for n ≥ 1 by 

and in general

Again,  is called the vector of ghost components of the Witt vector , and is usually denoted by .

We can use these polynomials to define the ring of universal Witt vectors over any commutative ring R in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring R).

Generating Functions
Witt also provided another approach using generating functions.

Definition
Let  be a Witt vector and define

For  let  denote the collection of subsets of  whose elements add up to . Then 

We can get the ghost components by taking the logarithmic derivative:

Sum
Now we can see  if . So that 

if  are the respective coefficients in the power series . Then 

Since  is a polynomial in  and likewise for , we can show by induction that  is a polynomial in

Product
If we set  then

But

.

Now 3-tuples  with  are in bijection with 3-tuples  with , via  ( is the least common multiple), our series becomes

So that 

where 's are polynomials of  So by similar induction, suppose 

 

then  can be solved as polynomials of

Ring schemes
The map taking a commutative ring R to the ring of Witt vectors over R (for a fixed prime p) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over  The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.

Similarly, the rings of truncated Witt vectors, and the rings of universal Witt vectors correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme.

Moreover, the functor taking the commutative ring  to the set  is represented by the affine space , and the ring structure on  makes  into a ring scheme denoted . From the construction of truncated Witt vectors, it follows that their associated ring scheme  is the scheme  with the unique ring structure such that the morphism  given by the Witt polynomials is a morphism of ring schemes.

Commutative unipotent algebraic groups

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group . The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

See also
Formal group
Artin–Hasse exponential
Necklace ring

References

 
 , section II.6
 
 

Category:Ring theory
Category:Algebraic groups
Category:Combinatorics on words