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.
We basically want to derive the ring -adic integers from the finite field with elements, , by some general construction.
A -adic integer is a sequence with ,such that if . They can be expanded as a power series in : , where the 's are usually taken from the set (The equation is happening in , with and all images from to ). Set-theoretically it is . But is not isomorphic to . If we denote , then the addition should instead be:
But we lack some properties of the coefficients to produce a general formula.
Luckily, there is an alternative subset of which can work as the coefficient set. This is the Teichmüller representatives of elements of . Without they form a subgroup of , identified with through the Teichmüller character . Note this is not additive, as the sum need not be a representative. Despite this, if in , then in . This is conceptually justified by if we denote .
They are explicitly calculated as roots of through Hensel lifting. For example, in , to calculate the representative of , you first find the unique solution of in with ; You get , then repeat it in , with conditions and ; This time it is , and so on. The existence of lift in each step is guaranteed by in every .
We can also write the representatives as . Note for every , there is exactly one representative, namely , with , so we can also expand every -adic integer as a power series in , with coefficients from the Teichmüller representatives.
Explicitly, if , then . Then you subtract and proceed similarly. Note the coefficients you get most probably differ from the 's, except the first one.
This time we have additional properties of the coefficients like , so we can make some changes to get a neat formula. Since the Teichmüller character is not additive, we don't have in . But it happens in , as the first congruence implies. So actually , thus . Since is divisible by , this resolves the -coefficient problem of and gives . Note this completely determines by the lift. Moreover, the indicates that the calculation can actually be done in , satisfying our basic aim.
Now for . It is already very cumbersome at this step. . As for , a single th power is not enough: actually we take . is not always divisible by , but that only happens when , in which case combined with similar monomials in would make a multiple of .
At this step, we see that we are actually working with something like
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
is called the ghost components of the Witt vector , and is usually denoted by .
Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring, called the ring of Witt vectors, such that
- the sum and product are given by polynomials with integral coefficients that do not depend on R, and
- Every Witt polynomial is a homomorphism from the ring of Witt vectors over R to R.
In other words, if
- and are given by polynomials with integral coefficients that do not depend on R, and
- , .
The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
- The Witt ring of any commutative ring R in which p is invertible is just isomorphic to RN (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 RN, 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, as is 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 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).
Later Witt orally stated another approach using generating functions.
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:
Now we can see if . So that if are respective coefficients in the power series for . Then . Since is a polynomial in and likely for , we can show by induction that is a polynomial in .
If we set then
Now 3-tuples with are in bijection with 3-tuples with , via ( is the Least common multiple), our series becomes
where s are polynomials of . So by similar induction, suppose , then can be solved as polynomials of .
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 Spec(Z). 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 Rn 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.
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