Witt vector

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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.

Motivation[edit]

We basically want to derive the ring p-adic integers \mathbb{Z}_p from the finite field with p elements, \mathbb{F}_p, by some general construction.

A p-adic integer is a sequence (n_0,n_1,...) with n_i\in\mathbb{Z}/p^{(n+1)}\mathbb{Z},such that n_i\equiv n_j\mod p^i if i<j. They can be expanded as a power series in p: a_0 + a_1 p^1 + a_2 p^2 + ..., where the a's are usually taken from the set \{0, 1, 2, ..., p-1\} (The equation is happening in \mathbb{Z}_p, with a_i and p^j all images from \mathbb{Z} to \mathbb{Z}_p). Set-theoretically it is \mathbb{F}_p. But \mathbb{Z}_p is not isomorphic to \prod_{\mathbb{N}}\mathbb{F}_p. If we denote a+b=c, then the addition should instead be:


c_0 \equiv a_0+b_0 \mod p

c_0+c_1 p\equiv a_0+a_1 p+b_0+b_1 p \mod p^2

c_0+c_1 p+c_2 p^2 \equiv a_0+a_1 p+a_2 p^2+b_0+b_1 p+b_2 p^2 \mod p^3

But we lack some properties of the coefficients to produce a general formula.

Luckily, there is an alternative subset of \mathbb{Z}_p which can work as the coefficient set. This is the Teichmüller representatives of elements of \mathbb{F}_p. Without 0 they form a subgroup of \mathbb{Z}_p^*, identified with \mathbb{F}_p^* through the Teichmüller character \omega:\mathbb{F}_p^*\rightarrow\mathbb{Z}_p^*. Note this is not additive, as the sum need not be a representative. Despite this, if \omega(k)=\omega(i)+\omega(j)\mod p in \mathbb{Z}_p, then i+j=k in \mathbb{F}_p. This is conceptually justified by m\circ \omega=\mathrm{id}_{\mathbb{F}_p} if we denote m:\mathbb{Z}_p\rightarrow\mathbb{Z}_p/p\mathbb{Z}_p\cong\mathbb{F}_p.

They are explicitly calculated as roots of x^{p-1}-1=0 through Hensel lifting. For example, in \mathbb{Z}_3, to calculate the representative of 2, you first find the unique solution of x^{2}-1=0 in \mathbb{Z}/9\mathbb{Z} with x\equiv 2\mod 3; You get 8, then repeat it in \mathbb{Z}/27\mathbb{Z}, with conditions x^{2}-1=0 and x\equiv 2\mod 9; This time it is 26, and so on. The existence of lift in each step is guaranteed by (x^{p-1}-1,(p-1)x^{p-2})=1 in every \mathbb{Z}/p^n\mathbb{Z}.

We can also write the representatives as a_{0} + a_{1} p^1 + a_{2} p^2 + .... Note for every j\in\{0, 1, 2, ..., p-1\}, there is exactly one representative, namely \omega(j), with a_{0}=j, so we can also expand every p-adic integer as a power series in p, with coefficients from the Teichmüller representatives.

Explicitly, if b=a_{0} + a_{1} p^1 + a_{2} p^2 + ..., then b-\omega(a_0)=a'_{1} p^1 + a'_{2} p^2 + .... Then you subtract \omega(a'_1)p and proceed similarly. Note the coefficients you get most probably differ from the a_i's, except the first one.

This time we have additional properties of the coefficients like a_i^p=a_i, so we can make some changes to get a neat formula. Since the Teichmüller character is not additive, we don't have c_0=a_0+b_0 in \mathbb{Z}_p. But it happens in \mathbb{F}_p, as the first congruence implies. So actually c_0^p\equiv (a_0+b_0)^p \mod p^2, thus c_0-a_0-b_0\equiv (a_0+b_0)^p-a_0-b_0\equiv \binom{p}{1} a_0^{p-1}b_0+...+ \binom{p}{1} a_0 b_0^{p-1} \mod p^2. Since \binom{p}{i} is divisible by p, this resolves the p-coefficient problem of c_1 and gives c_1\equiv a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1}\mod p. Note this completely determines c_1 by the lift. Moreover, the \mod p indicates that the calculation can actually be done in \mathbb{F}_p, satisfying our basic aim.

Now for c_2. It is already very cumbersome at this step. c_1=c_1^p \equiv (a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1})^p\mod p. As for c_0, a single pth power is not enough: actually we take c_0=c_0^{p^2}\equiv(a_0+b_0)^{p^2}. \binom{p^2}{i} is not always divisible by p^2, but that only happens when i=pd, in which case a^ib^{p^2-i}=a^db^{p-d} combined with similar monomials in c_1^p would make a multiple of p^2.

At this step, we see that we are actually working with something like


c_0 \equiv a_0+b_0 \mod p

c_0^p+c_1 p\equiv a_0^p+a_1 p+b_0^p+b_1 p \mod p^2

c_0^{p^2}+c_1^p p+c_2 p^2 \equiv a_0^{p^2}+a_1^p p+a_2 p^2+b_0^{p^2}+b_1^p p+b_2 p^2 \mod p^3

This motivates the definition of Witt vectors.

Construction of Witt rings[edit]

Fix a prime number p. A Witt vector over a commutative ring R is a sequence : (X_0,X_1,X_2,...) of elements of R. Define the Witt polynomials W_i by

  1.  W_0=X_0\,
  2.  W_1=X_0^p+pX_1
  3.  W_2=X_0^{p^2}+pX_1^p+p^2X_2

and in general

 W_n=\sum_ip^iX_i^{p^{n-i}}.

 (W_0,W_1,W_2,...) is called the ghost components of the Witt vector (X_0,X_1,X_2,...), and is usually denoted by  (X^{(0)},X^{(1)},X^{(2)},...).

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

  •  (X+Y)_i and  (XY)_i are given by polynomials with integral coefficients that do not depend on R, and
  •  X^{(i)}+Y^{(i)}=(X+Y)^{(i)},  X^{(i)}Y^{(i)}=(XY)^{(i)}.

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

  • (X_0,X_1,...)+(Y_0,Y_1,...)=(X_0+Y_0,X_1+Y_1+(X_0^p+Y_0^p-(X_0+Y_0)^p)/p,...)
  • (X_0,X_1,...)\times(Y_0,Y_1,...)=(X_0 Y_0,X_0^p Y_1+X_1 Y_0^p+p X_1 Y_1,...).

Examples[edit]

  • 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[edit]

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

  1.  W_1=X_1\,
  2.  W_2=X_1^2+2X_2
  3.  W_3=X_1^3+3X_3
  4.  W_4=X_1^{4}+2X_2^2+4X_4

and in general

 W_n=\sum_{d|n}dX_d^{n/d}.

Again, (W_1,W_2,W_3,...) is called the ghost components of the Witt vector (X_1,X_2,X_3,...), and is usually denoted by  (X^{(1)},X^{(2)},X^{(3)},...).

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[edit]

Later Witt orally stated another approach using generating functions.[1]

Definition[edit]

Let X be a Witt vector and define

f_X(t)=\prod_{n\ge 1}(1-X_n t^n)=\sum_{n\ge 0}A_n t^n

For n\ge 1 let \mathcal{S}_n denote the collection of subsets of \{1,2,...,n\} whose elements add up to n. Then A_n=\sum_{S\in\mathcal{S}}(-1)^{|S|}\sum_{i\in S}{X_i}.

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

\frac{d}{dt}\log f_X(t)=\sum_{n\ge 1}\frac{d}{dt}(1-X_n t^n)=-\sum_{n\ge 1}\sum_{d\ge 1}\frac{X_n^d t^{nd}}{d}=-\sum_{m\ge 1}\frac{\sum_{d|m}\frac{m}{d}X_{\frac{m}{d}}^d}{m}t^m=-\sum_{m\ge 1}\frac{X^{(m)}t^m}{m}

Sum[edit]

Now we can see f_{Z}(t)=f_X(t) f_Y(t) if Z=X+Y. So that C_n=\sum_{0\le i\le n}A_n B_{n-i} if A_n,B_n,C_n are respective coefficients in the power series for f_X(t),f_Y(t),f_Z(t). Then Z_n=\sum_{0\le i\le n}A_n B_{n-i}-\sum_{S\in\mathcal{S},S\ne\{n\}}(-1)^{|S|}\sum_{i\in S}{Z_i}. Since A_n is a polynomial in X_1,...,X_n and likely for B_n, we can show by induction that Z_n is a polynomial in X_1,...,X_n,Y_1,...,Y_n.

Product[edit]

If we set W=XY then

\frac{d}{dt}\log f_W(t)=-\sum_{m\ge 1}\frac{X^{(m)}Y^{(m)}t^m}{m}

But

\sum_{m\ge 1}\frac{X^{(m)}Y^{(m)}}{m}t^m=\sum_{m\ge 1}\frac{\sum_{d|m}d X_d^{m/d}\sum_{e|m}e Y_e^{m/e}}{m}t^m

Now 3-tuples {m,d,e} with m\in\mathbb{Z}^+,d|m,e|m are in bijection with 3-tuples {d,e,n} with d,e,n\in\mathbb{Z}^+, via n=m/[d,e] ([d,e] is the Least common multiple), our series becomes

\sum_{d,e\ge 1}\frac{\frac{d e}{ [d,e]}\sum_{n\ge 1} (X_d^{ [d,e]/d } Y_e^{ [d,e]/e } t^{ [d,e] })^n}{n}

So that

f_W(t)=\prod_{d,e\ge 1}(1-X_d^{[d,e]/d}Y_e^{[d,e]/e} t^{[d,e]})^{d e/[d,e]}=\sum_{n\ge 0}D_n t^n

where D_ns are polynomials of X_1,...,X_n,Y_1,...,Y_n. So by similar induction, suppose f_W(t)=\prod_{n\ge 1}(1-W_n t^n), then W_n can be solved as polynomials of X_1,...,X_n,Y_1,...,Y_n.

Ring schemes[edit]

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 R to the set R^n is represented by the affine space \mathbb{A}_{\mathbb{Z}}^n, and the ring structure on Rn makes \mathbb{A}_{\mathbb{Z}}^n into a ring scheme denoted \underline{\mathcal{O}}^n. From the construction of truncated Witt vectors it follows that their associated ring scheme \mathbb{W}_n is the scheme \mathbb{A}_{\mathbb{Z}}^n with the unique ring structure such that the morphism \mathbb{W}_n\rightarrow \underline{\mathcal{O}}^n given by the Witt polynomials is a morphism of ring schemes.

Commutative unipotent algebraic groups[edit]

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 G_a. 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[edit]

References[edit]

  1. ^ Lang, Serge (September 19, 2005). "Chapter VI: Galois Theory". Algebra (3rd ed.). Springer. p. 330. ISBN 978-0-387-95385-4.