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 $\Delta \subseteq k^{\times }/(k^{\times })^{n}$, where $\Delta$ corresponds to $K=k({\sqrt[{n}]{\Delta }})$.

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 $x^{p}=1$. Because raising to the pth power is the Frobenius homomorphism, this equation may be rewritten as $(x-1)^{p}=0$, and therefore $x=1$. 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 $x^{p}-x-a$. 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 W_n(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

$F(x)-x-a,$

where $a\in W_{n}(k)$. To complete the analogy with Kummer theory, define $\wp$ to be the operator $x\mapsto F(x)-x$. Then the degree pⁿ extensions of k are in bijective correspondence with cyclic subgroups $\Delta \subseteq W_{n}(k)/\wp (W_{n}(k))$ of order pⁿ, where $\Delta$ corresponds to the field $k(\wp ^{-1}(\Delta ))$.

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 Oswald 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 $n_{i}\equiv n_{j}\!\!\!\!\mod p^{i+1}$ 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 $x=x_{0}+x_{1}p^{1}+x_{2}p^{2}+\cdots$, one takes the difference $x-\omega (x_{0})=x'_{1}p^{1}+x'_{2}p^{2}+\cdots$, leaving a value divisible by $p$. Hence, $x-\omega (x_{0})=0\mod p$. The process is then repeated, subtracting $\omega (x'_{1})p$ and proceed likewise. This yields a sequence of congruences

$x\equiv \omega (x_{0})\mod p,$

$x\equiv \omega (x_{0})+\omega (x'_{1})p\mod p^{2},$

$...$;

So that $x\equiv \Sigma _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}\mod p^{i+1}$

and $i'>i\Rightarrow \Sigma _{j=0}^{i'}\omega ({\bar {x}}_{j})p^{j}\equiv \Sigma _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}\mod p^{i+1}$

for ${\bar {x}}_{i}:=m\left({\frac {x-\Sigma _{j=0}^{i-1}\omega ({\bar {x}}_{j})p^{j}}{p^{i}}}\right)$.

Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than $\{0,...,p-1\}$. It is clear that $\Sigma _{j=0}^{\infty }\omega ({\bar {x}}_{j})p^{j}=x$, since $p^{i+1}\mid x-\Sigma _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}$ for all $i$ as $i\to \infty$, so the difference tends to 0 with respect to the p-adic metric. 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 $\omega ({\bar {x}}_{i})^{p}=\omega ({\bar {x}}_{i})$, 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

$c_{1}=c_{1}^{p}\equiv \left(a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}\right)^{p}\!\!\mod p.$

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: $(X_{0},X_{1},X_{2},\ldots )$ 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. $W_{2}=X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}$

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,

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

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

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

Witt also provided another approach using generating functions.

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

$A_{n}=\sum _{S\in {\mathcal {S}}_{n}}(-1)^{|S|}\sum _{i\in S}{X_{i}}$.

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 $X_{1},\ldots ,X_{n}$ and likely for $B_{n}$, we can show by induction that $Z_{n}$ is a polynomial in $X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}$.

Product

If we set $W=XY$ then

$-t{\frac {d}{dt}}\log f_{W}(t)=-\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}$.

But

$\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}\sum _{e|m}eY_{e}^{m/e}t^{m}$.

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

$\sum _{d,e\geq 1}de\sum _{n\geq 1}(X_{d}^{[d,e]/d}Y_{e}^{[d,e]/e}t^{[d,e]})^{n}=-t{\frac {d}{dt}}\log \prod _{d,e\geq 1}(1-X_{d}^{[d,e]/d}Y_{e}^{[d,e]/e}t^{[d,e]})^{de/[d,e]}$

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.

See also

• Formal group
• Artin–Hasse exponential
• Necklace ring

References

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