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.

Motivation

Any $p$-adic integer (an element of $\mathbb{Z}_p$) can be written as a power series $a_0 + a_1 p^1 + a_2 p^2 + \cdots$, where the $a$'s are usually taken from the set $\{0, 1, 2, ..., p-1\}$. However, it is hard to figure out an algebraic expression for addition and multiplication, as one faces the problem of carrying. Luckily, this set of representatives is not the only possible choice, and Teichmüller suggested an alternative set consisting of 0 together with the $p-1$st roots of $1$: in other words, the $p$ roots of

$x^p - x = 0$ in $\mathbb{Z}_p$.

These Teichmüller representatives can be identified with the elements of the finite field $\mathbb{F}_p$ of order $p$ (by taking residues modulo $p$), and elements of $\mathbb{F}_p^\times$ are taken to their representatives by the Teichmüller character $\omega:\mathbb{F}_p^\times \rightarrow \mathbb{Z}_p^\times$. This identifies the set of $p$-adic integers with infinite sequences of elements of $\omega(\mathbb{F}_p^\times) \cup \{0\}$.

We now have the following problem: given two infinite sequences of elements of $\omega(\mathbb{F}_p^\times) \cup \{0\}$, describe their sum and product as $p$-adic integers explicitly. This problem was solved by Witt using Witt vectors.

Details

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.

The ring $\mathbb Z_p$ of $p$-adic integers consists of the sequences $(n_0,n_1,...)$ with $n_i\in\mathbb{Z}/p^{i+1}\mathbb{Z}$,such that $n_i\equiv n_j\mod p^i$ if $i\le j$. (It is a projective limit.) Its elements can be expanded as (formal) power series $a_0 + a_1 p^1 + a_2 p^2 + \cdots$ in $p$, where the $a_i$'s are usually taken from the set $\{0, 1, 2, ..., p-1\}$. (The power series usually do not converge in $\mathbb R$, but do converge in $\mathbb{Z}_p$, with $a_i$ and $p^j$ being identified with their images under $\mathbb{Z} \to \mathbb{Z}_p$.) Set-theoretically, $\mathbb Z_p$ is just $\prod_{\mathbb N} \mathbb{F}_p$; but the two sets are not isomorphic as rings. If we denote $a+b$ by $c$, 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 set of 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 that $\omega$ 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 $m:\mathbb{Z}_p\rightarrow\mathbb{Z}_p/p\mathbb{Z}_p\cong\mathbb{F}_p$.

Teichmüller representatives 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 $x\equiv 8\mod 9$; 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 modulo $p$, 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 $p$th 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

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

• The Witt ring of any commutative ring R in which p is invertible is just isomorphic to R^N (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 R^N, 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 W_n 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

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

Definition

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

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

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 $D_n$s 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

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

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

• Formal group
• Artin–Hasse exponential

References

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

• Dolgachev, I.V. (2001), "Witt vector", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, ISBN 978-0-444-53257-2, MR 2553661
• Mumford, David, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237, section II.6
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics 117, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96648-9, MR 918564
• Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pⁿ. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pⁿ", Journal für Reine und Angewandte Mathematik (in German) 176: 126–140, doi:10.1515/crll.1937.176.126
• Greenberg, M. J. (1969), Lectures on Forms in Many Variables, New York and Amsterdam, Benjamin, MR 241358, ASIN: B0006BX17M