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 $\mathbb {Z} _{p}$ , not to be confused with $\mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}$ ) 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 $\{0,1,2,\cdots ,p-1\}=\mathbb {F} _{p}$ . However, it is hard to provide an algebraic expression for addition and multiplication, using this representation for the p-adic integers, as one faces the problem of carrying. However, this set of representative coefficients (that is, taking the coefficients $a_{i}$ from $\mathbb {F} _{p}$ ) is not the only possible choice, and Teichmüller suggested an alternative set of coefficients, taken from $\mathbb {Z} _{p}$ (that is, each $a_{i}\in \mathbb {Z} _{p}$ ) such that expressions for addition and multiplication can be written in closed form. These coefficients $a_{i}$ consist of 0 together with the $p-1$ th roots of unity; that is, the roots of $x^{p}-x=0$ in $\mathbb {Z} _{p},$ so that $a_{i}=a_{i}^{p}.$

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.

Detailed motivational sketch

The below derives the ring of $p$ -adic integers $\mathbb {Z} _{p}$ from the finite field with $p$ elements, $\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z}$ , using a construction which naturally generalizes to the Witt vector construction.

The ring ${\mathbb Z}_{p}$ of $p$ -adic integers can be understood as the projective limit of $\mathbb {Z} /p^{i}\mathbb {Z} ,$ taking $i\to \infty$ . Specifically, it consists of the sequences $(n_{0},n_{1},\cdots )$ with $n_{i}\in \mathbb {Z} /p^{i+1}\mathbb {Z}$ , such that $n_{i}\equiv n_{j}\!\!\!\!\mod p^{i}$ if $i\leq j.$ That is, the next element of the sequence equals the last, modulo one more power of p; this gives the projection defining the inverse limit.

The elements of ${\mathbb Z}_{p}$ 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 $\{0,1,2,\cdots ,p-1\}$ . Of course, this power series usually will not converge in $\mathbb {R}$ using the standard metric on the reals, but it will converge in $\mathbb {Z} _{p}$ , with the p-adic metric. Viewed as a sequence, ${\mathbb Z}_{p}$ is just $\prod _{\mathbb {N} }\mathbb {F} _{p}$ , if one forgets the ring structure. What follows is a sketch of how a ring structure can be provided for the sequence.

Letting $a+b$ be denoted by $c$ , one might consider the following definition for addition:

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}

However, this lacks several properties needed to produce a general formula; most notably, one does not have that $c_{i}=c_{i}^{p}.$

There is an alternative subset of $\mathbb {Z} _{p}$ which can be used 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 $\mathbb {Z} _{5}$ , to calculate the representative of $2$ , one starts by finding the unique solution of $x^{4}-1=0$ in $\mathbb {Z} /25\mathbb {Z}$ with $x\equiv 2\!\!\!\!\mod 5$ ; one gets $7$ . Repeat this in $\mathbb {Z} /125\mathbb {Z}$ , with the conditions $x^{4}-1=0$ and $x\equiv 7\!\!\!\!\mod 25$ gives $57$ , and so on; the resulting Teichmüller representative is $(2,7,57,\cdots ).$ The existence of a lift in each step is guaranteed by the greatest common divisor $(x^{{p-1}}-1,(p-1)x^{{p-2}})=1$ in every ${\mathbb {Z}}/p^{n}{\mathbb {Z}}$ .

Note for every $j\in \{0,1,2,\cdots ,p-1\}$ , there is exactly one representative, namely $\omega (j)$ , with $a_{{0}}=j$ . Because of this one-to-one correspondence, one can expand every $p$ -adic integer as a power series in $p$ , with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as $\omega (t_{0})=t_{0}+t_{1}p^{1}+t_{2}p^{2}+\cdots$ . Then, if one has some arbitrary p-adic integer of the form $b=a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots$ , one takes the difference $b-\omega (a_{0})=a'_{1}p^{1}+a'_{2}p^{2}+\cdots$ , leaving a value divisible by $p$ . The process is then repeated, subtracting $\omega (a'_{1})p$ and proceed likewise. The resulting coefficients will typically differ from the $a_{i}$ 's modulo $p^{i}$ , except the first one.

This transformed sequence of coefficients has an additional property, namely that $a_{i}^{p}=a_{i}$ , which the original sequence did not have. This can be used to describe addition, as follows. Since the Teichmüller character is not additive, it is not true that $c_{0}=a_{0}+b_{0}$ in $\mathbb {Z} _{p}$ . But it does hold in $\mathbb {F} _{p}$ , as the first congruence implies. In particular, $c_{0}^{p}\equiv (a_{0}+b_{0})^{p}\!\!\!\!\mod p^{2}$ , and 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}+\cdots +{\binom {p}{p-1}}a_{0}b_{0}^{p-1}\!\!\!\!\mod p^{2}.

Since the binomial coefficient ${\binom {p}{i}}$ is divisible by $p$ , this 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}-\cdots -a_{0}b_{0}^{p-1}\!\!\!\!\mod p.

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 the basic aim of defining a simple additive structure.

Now for $c_{2}$ . It is already very cumbersome at this step. Write

c_{1}=c_{1}^{p}\equiv \left(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}\right)^{p}\!\!\mod p.

Just as for $c_{0}$ , a single $p$ th power is not enough: one must take

c_{0}=c_{0}^{p^{2}}\equiv (a_{0}+b_{0})^{p^{2}}.

However, ${\binom {p^{2}}{i}}$ is not in general divisible by $p^{2}$ , but it is divisible when $i=pd$ , in which case $a^{i}b^{{p^{2}-i}}=a^{d}b^{{p-d}}$ combined with similar monomials in $c_{1}^{p}$ will make a multiple of $p^{2}$ .

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

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

$W_{0}=X_{0}\,$
$W_{1}=X_{0}^{p}+pX_{1}$
$W_{2}=X_{0}^{{p^{2}}}+pX_{1}^{p}+p^{2}X_{2}$

and in general

W_{n}=\sum _{i}p^{i}X_{i}^{{p^{{n-i}}}}.

The $W_{n}$ are called the ghost components of the Witt vector $(X_{0},X_{1},X_{2},\cdots )$ , and are usually denoted by $X^{(n)}.$

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
$X^{{(i)}}+Y^{{(i)}}=(X+Y)^{{(i)}}$ and $X^{{(i)}}Y^{{(i)}}=(XY)^{{(i)}}$ .

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

$(X_{0},X_{1},...)+(Y_{0},Y_{1},...)=(X_{0}+Y_{0},\;X_{1}+Y_{1}+(X_{0}^{p}+Y_{0}^{p}-(X_{0}+Y_{0})^{p})/p,\;\cdots )$
$(X_{0},X_{1},...)\times (Y_{0},Y_{1},...)=(X_{0}Y_{0},\;X_{0}^{p}Y_{1}+X_{1}Y_{0}^{p}+pX_{1}Y_{1},\;\cdots )$ .

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

$W_{1}=X_{1}\,$
$W_{2}=X_{1}^{2}+2X_{2}$
$W_{3}=X_{1}^{3}+3X_{3}$
$W_{4}=X_{1}^{{4}}+2X_{2}^{2}+4X_{4}$

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

Witt also provided another approach using generating functions.^[1]

Definition

Let $X$ be a Witt vector and define

f_{X}(t)=\prod _{{n\geq 1}}(1-X_{n}t^{n})=\sum _{{n\geq 0}}A_{n}t^{n}

For $n\geq 1$ 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:

{\begin{aligned}-t{\frac {d}{dt}}\log f_{X}(t)=&-t{\frac {d}{dt}}\sum _{n\geq 1}\log(1-X_{n}t^{n})=t{\frac {d}{dt}}\sum _{n\geq 1}\sum _{d\geq 1}{\frac {X_{n}^{d}t^{nd}}{d}}\\=&\sum _{n\geq 1}\sum _{d\geq 1}nX_{n}^{d}t^{nd}=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}t^{m}=\sum _{m\geq 1}X^{(m)}t^{m}\end{aligned}}

Sum

Now we can see $f_{{Z}}(t)=f_{X}(t)f_{Y}(t)$ if $Z=X+Y$ . So that

C_{n}=\sum _{{0\leq i\leq n}}A_{n}B_{{n-i}}

if $A_{n},B_{n},C_{n}$ are the respective coefficients in the power series $f_{X}(t),f_{Y}(t),f_{Z}(t)$ . Then

Z_{n}=\sum _{{0\leq i\leq n}}A_{n}B_{{n-i}}-\sum _{{S\in {\mathcal {S}},S\neq \{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\geq 1}}{\frac {X^{{(m)}}Y^{{(m)}}t^{m}}{m}}

But

\sum _{{m\geq 1}}{\frac {X^{{(m)}}Y^{{(m)}}}{m}}t^{m}=\sum _{{m\geq 1}}{\frac {\sum _{{d|m}}dX_{d}^{{m/d}}\sum _{{e|m}}eY_{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\geq 1}}{\frac {{\frac {de}{[d,e]}}\sum _{{n\geq 1}}(X_{d}^{{[d,e]/d}}Y_{e}^{{[d,e]/e}}t^{{[d,e]}})^{n}}{n}}

So that

f_{W}(t)=\prod _{{d,e\geq 1}}(1-X_{d}^{{[d,e]/d}}Y_{e}^{{[d,e]/e}}t^{{[d,e]}})^{{de/[d,e]}}=\sum _{{n\geq 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\geq 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.

References

↑ 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

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