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 ${\displaystyle W(\mathbb {F} _{p})}$ over the finite field of prime order p is isomorphic to ${\displaystyle \mathbb {Z} _{p}}$, the ring of p-adic integers. They have a highly non-intuitive structure^[1] upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers.

The main idea^[1] behind Witt vectors is that instead of using the standard p-adic expansion

${\displaystyle a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots }$

to represent an element in ${\displaystyle \mathbb {Z} _{p}}$, we can instead consider an expansion using the Teichmüller character

${\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}$

which sends each element in the solution set of ${\displaystyle x^{p-1}-1}$ in ${\displaystyle \mathbb {F} _{p}}$ to an element in the solution set of ${\displaystyle x^{p-1}-1}$ in ${\displaystyle \mathbb {Z} _{p}}$. That is, we expand out elements in ${\displaystyle \mathbb {Z} _{p}}$ in terms of roots of unity instead of as profinite elements in ${\displaystyle \prod \mathbb {F} _{p}}$. We can then express a p-adic integer as an infinite sum

${\displaystyle \omega (a)=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots }$

which gives a Witt vector

${\displaystyle (\omega (a_{0}),\omega (a_{1}),\omega (a_{2}),\ldots )}$

Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give ${\displaystyle W(\mathbb {F} _{p})}$ an additive and multiplicative structure such that ${\displaystyle \omega }$ induces a commutative ring homomorphism.

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 ${\displaystyle k}$ be a field containing a primitive ${\displaystyle n}$-th root of unity. Kummer theory classifies degree ${\displaystyle n}$ cyclic field extensions ${\displaystyle K}$ of ${\displaystyle k}$. Such fields are in bijection with order ${\displaystyle n}$ cyclic groups ${\displaystyle \Delta \subseteq k^{\times }/(k^{\times })^{n}}$, where ${\displaystyle \Delta }$ corresponds to ${\displaystyle K=k({\sqrt[{n}]{\Delta }}\,)}$.

But suppose that ${\displaystyle k}$ has characteristic ${\displaystyle p}$. The problem of studying degree ${\displaystyle p}$ extensions of ${\displaystyle k}$, or more generally degree ${\displaystyle p^{n}}$ extensions, may appear superficially similar to Kummer theory. However, in this situation, ${\displaystyle k}$ cannot contain a primitive ${\displaystyle p}$-th root of unity. Indeed, if ${\displaystyle x}$ is a ${\displaystyle p}$-th root of unity in ${\displaystyle k}$, then it satisfies ${\displaystyle x^{p}=1}$. But consider the expression ${\displaystyle (x-1)^{p}=0}$. By expanding using binomial coefficients we see that the operation of raising to the ${\displaystyle p}$-th power, known here as the Frobenius homomorphism, introduces the factor ${\displaystyle p}$ to every coefficient except the first and the last, and so modulo ${\displaystyle p}$ these equations are the same. Therefore ${\displaystyle x=1}$. Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.

The case where the characteristic divides the degree is today 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.^[2] 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 ${\displaystyle p}$ extensions of a field ${\displaystyle k}$ of characteristic ${\displaystyle p}$ were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form ${\displaystyle x^{p}-x-a.}$ By repeating their construction, they described degree ${\displaystyle p^{2}}$ extensions. Abraham Adrian Albert used this idea to describe degree ${\displaystyle p^{n}}$ extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.^[3]

Schmid^[4] generalized further to non-commutative cyclic algebras of degree ${\displaystyle p^{n}}$. In the process of doing so, certain polynomials related to the addition of ${\displaystyle p}$-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree ${\displaystyle p^{n}}$ field extensions and cyclic algebras. Specifically, he introduced a ring now called ${\displaystyle W_{n}(k)}$, the ring of ${\displaystyle n}$-truncated ${\displaystyle p}$-typical Witt vectors. This ring has ${\displaystyle k}$ as a quotient, and it comes with an operator ${\displaystyle F}$ which is called the Frobenius operator because it reduces to the Frobenius operator on ${\displaystyle k}$. Witt observes that the degree ${\displaystyle p^{n}}$ analog of Artin–Schreier polynomials is

${\displaystyle F(x)-x-a,}$

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

Any ${\displaystyle p}$-adic integer (an element of ${\displaystyle \mathbb {Z} _{p}}$, not to be confused with ${\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}$) can be written as a power series ${\displaystyle a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots }$, where the ${\displaystyle a_{i}}$ are usually taken from the integer interval ${\displaystyle [0,p-1]=\{0,1,2,\ldots ,p-1\}}$. It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients ${\displaystyle a_{i}\in [0,p-1]}$ is only one of many choices, and Hensel himself (the creator of ${\displaystyle p}$-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number ${\displaystyle 0}$ together with the ${\displaystyle (p-1)^{\text{st}}}$ roots of unity; that is, the solutions of ${\displaystyle x^{p}-x=0}$ in ${\displaystyle \mathbb {Z} _{p}}$, so that ${\displaystyle a_{i}=a_{i}^{p}}$. This choice extends naturally to ring extensions of ${\displaystyle \mathbb {Z} _{p}}$ in which the residue field is enlarged to ${\displaystyle \mathbb {F} _{q}}$ with ${\displaystyle q=p^{f}}$, some power of ${\displaystyle p}$. Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the ${\displaystyle q}$ solutions in the field to ${\displaystyle x^{q}-x=0}$. Call the field ${\displaystyle \mathbb {Z} _{p}(\eta )}$, with ${\displaystyle \eta }$ an appropriate primitive ${\displaystyle (q-1)^{\text{th}}}$ root of unity (over ${\displaystyle \mathbb {Z} _{p}}$). The representatives are then ${\displaystyle 0}$ and ${\displaystyle \eta ^{i}}$ for ${\displaystyle 0\leq i\leq q-2}$. Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works, Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field ${\displaystyle \mathbb {F} _{q}}$ of order ${\displaystyle q}$ by taking residues modulo ${\displaystyle p}$ in ${\displaystyle \mathbb {Z} _{p}(\eta )}$, and elements of ${\displaystyle \mathbb {F} _{q}^{\times }}$ are taken to their representatives by the Teichmüller character ${\displaystyle \omega :\mathbb {F} _{q}^{\times }\to \mathbb {Z} _{p}(\eta )^{\times }}$. This operation identifies the set of integers in ${\displaystyle \mathbb {Z} _{p}(\eta )}$ with infinite sequences of elements of ${\displaystyle \omega (\mathbb {F} _{q}^{\times })\cup \{0\}}$.

Taking those representatives, the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: ${\displaystyle q=p}$): given two infinite sequences of elements of ${\displaystyle \omega (\mathbb {F} _{p}^{\times })\cup \{0\},}$ describe their sum and product as p-adic integers explicitly. This problem was solved by Witt using Witt vectors.

We derive the ring of ${\displaystyle p}$-adic integers ${\displaystyle \mathbb {Z} _{p}}$ from the finite field ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ using a construction which naturally generalizes to the Witt vector construction.

The ring ${\displaystyle \mathbb {Z} _{p}}$ of p-adic integers can be understood as the inverse limit of the rings ${\displaystyle \mathbb {Z} /p^{i}\mathbb {Z} }$ taken along the obvious projections. Specifically, it consists of the sequences ${\displaystyle (n_{0},n_{1},\ldots )}$ with ${\displaystyle n_{i}\in \mathbb {Z} /p^{i+1}\mathbb {Z} ,}$ such that ${\displaystyle n_{j}\equiv n_{i}{\bmod {p}}^{i+1}}$ for ${\displaystyle j\geq i.}$ That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections ${\displaystyle \mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} .}$

The elements of ${\displaystyle \mathbb {Z} _{p}}$ can be expanded as (formal) power series in ${\displaystyle p}$

${\displaystyle a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots ,}$

where the coefficients ${\displaystyle a_{i}}$ are taken from the integer interval ${\displaystyle [0,p-1]=\{0,1,\ldots ,p-1\}.}$ Of course, this power series usually will not converge in ${\displaystyle \mathbb {R} }$ using the standard metric on the reals, but it will converge in ${\displaystyle \mathbb {Z} _{p},}$ with the p-adic metric. We will sketch a method of defining ring operations for such power series.

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

${\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}+c_{1}p&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p&&{\bmod {p}}^{2}\\c_{0}+c_{1}p+c_{2}p^{2}&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p+(a_{2}+b_{2})p^{2}&&{\bmod {p}}^{3}\end{aligned}}}$

and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set ${\displaystyle [0,p-1].}$

There is a better coefficient subset of ${\displaystyle \mathbb {Z} _{p}}$ which does yield closed formulas, the Teichmüller representatives: zero together with the ${\displaystyle (p-1)^{\text{th}}}$ roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives ${\displaystyle [0,p-1]}$) as roots of ${\displaystyle x^{p-1}-1=0}$ through Hensel lifting, the p-adic version of Newton's method. For example, in ${\displaystyle \mathbb {Z} _{5},}$ to calculate the representative of ${\displaystyle 2,}$ one starts by finding the unique solution of ${\displaystyle x^{4}-1=0}$ in ${\displaystyle \mathbb {Z} /25\mathbb {Z} }$ with ${\displaystyle x\equiv 2{\bmod {5}}}$; one gets ${\displaystyle 7.}$ Repeating this in ${\displaystyle \mathbb {Z} /125\mathbb {Z} ,}$ with the conditions ${\displaystyle x^{4}-1=0}$ and ${\displaystyle x\equiv 7{\bmod {2}}5}$, gives ${\displaystyle 57,}$ and so on; the resulting Teichmüller representative of ${\displaystyle 2}$, denoted ${\displaystyle \omega (2)}$, is the sequence

${\displaystyle \omega (2)=(2,7,57,\ldots )\in W(\mathbb {F} _{5}).}$

The existence of a lift in each step is guaranteed by the greatest common divisor ${\displaystyle (x^{p-1}-1,(p-1)x^{p-2})=1}$ in every ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}$

This algorithm shows that for every ${\displaystyle j\in [0,p-1]}$, there is exactly one Teichmüller representative with ${\displaystyle a_{0}=j}$, which we denote ${\displaystyle \omega (j).}$ Indeed, this defines the Teichmüller character ${\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}$ as a (multiplicative) group homomorphism, which moreover satisfies ${\displaystyle m\circ \omega =\mathrm {id} _{\mathbb {F} _{p}}}$ if we let ${\displaystyle m:\mathbb {Z} _{p}\to \mathbb {Z} _{p}/p\mathbb {Z} _{p}\cong \mathbb {F} _{p}}$ denote the canonical projection. Note however that ${\displaystyle \omega }$ is not additive, as the sum need not be a representative. Despite this, if ${\displaystyle \omega (k)\equiv \omega (i)+\omega (j){\bmod {p}}}$ in ${\displaystyle \mathbb {Z} _{p},}$ then ${\displaystyle i+j=k}$ in ${\displaystyle \mathbb {F} _{p}.}$

Because of this one-to-one correspondence given by ${\displaystyle \omega }$, 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 ${\displaystyle \omega (t_{0})=t_{0}+t_{1}p^{1}+t_{2}p^{2}+\cdots .}$ Then, if one has some arbitrary p-adic integer of the form ${\displaystyle x=x_{0}+x_{1}p^{1}+x_{2}p^{2}+\cdots ,}$ one takes the difference ${\displaystyle x-\omega (x_{0})=x'_{1}p^{1}+x'_{2}p^{2}+\cdots ,}$ leaving a value divisible by ${\displaystyle p}$. Hence, ${\displaystyle x-\omega (x_{0})=0{\bmod {p}}}$. The process is then repeated, subtracting ${\displaystyle \omega (x'_{1})p}$ and proceed likewise. This yields a sequence of congruences

${\displaystyle {\begin{aligned}x&\equiv \omega (x_{0})&&{\bmod {p}}\\x&\equiv \omega (x_{0})+\omega (x'_{1})p&&{\bmod {p}}^{2}\\&\cdots \end{aligned}}}$

So that

${\displaystyle x\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}$

and ${\displaystyle i'>i}$ implies:

${\displaystyle \sum _{j=0}^{i'}\omega ({\bar {x}}_{j})p^{j}\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}$

for

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

Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than ${\displaystyle \{0,\ldots ,p-1\}}$. It is clear that

${\displaystyle \sum _{j=0}^{\infty }\omega ({\bar {x}}_{j})p^{j}=x,}$

since

${\displaystyle p^{i+1}\mid x-\sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}}$

for all ${\displaystyle i}$ as ${\displaystyle i\to \infty ,}$ so the difference tends to 0 with respect to the p-adic metric. The resulting coefficients will typically differ from the ${\displaystyle a_{i}}$ modulo ${\displaystyle p^{i}}$ except the first one.

The Teichmüller coefficients have the key additional property that ${\displaystyle \omega ({\bar {x}}_{i})^{p}=\omega ({\bar {x}}_{i}),}$ which is missing for the numbers in ${\displaystyle [0,p-1]}$. This can be used to describe addition, as follows. Consider the equation ${\textstyle c=a+b}$ in ${\textstyle \mathbb {Z} _{p}}$ and let the coefficients ${\textstyle a_{i},b_{i},c_{i}\in \mathbb {Z} _{p}}$ now be as in the Teichmüller expansion. Since the Teichmüller character is not additive, ${\displaystyle c_{0}=a_{0}+b_{0}}$ is not true in ${\displaystyle \mathbb {Z} _{p}}$. But it holds in ${\displaystyle \mathbb {F} _{p},}$ as the first congruence implies. In particular,

${\displaystyle c_{0}^{p}\equiv (a_{0}+b_{0})^{p}{\bmod {p}}^{2},}$

and thus

${\displaystyle c_{0}-a_{0}-b_{0}\equiv (a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv {\binom {p}{1}}a_{0}^{p-1}b_{0}+\cdots +{\binom {p}{p-1}}a_{0}b_{0}^{p-1}{\bmod {p}}^{2}.}$

Since the binomial coefficient ${\displaystyle {\binom {p}{i}}}$ is divisible by ${\displaystyle p}$, this gives

${\displaystyle c_{1}\equiv a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}{\bmod {p}}.}$

This completely determines ${\displaystyle c_{1}}$ by the lift. Moreover, the congruence modulo ${\displaystyle p}$ indicates that the calculation can actually be done in ${\displaystyle \mathbb {F} _{p},}$ satisfying the basic aim of defining a simple additive structure.

For ${\displaystyle c_{2}}$ this step is already very cumbersome. Write

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

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

${\displaystyle c_{0}=c_{0}^{p^{2}}\equiv (a_{0}+b_{0})^{p^{2}}{\bmod {p}}^{3}.}$

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

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

${\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}^{p}+c_{1}p&\equiv a_{0}^{p}+a_{1}p+b_{0}^{p}+b_{1}p&&{\bmod {p}}^{2}\\c_{0}^{p^{2}}+c_{1}^{p}p+c_{2}p^{2}&\equiv a_{0}^{p^{2}}+a_{1}^{p}p+a_{2}p^{2}+b_{0}^{p^{2}}+b_{1}^{p}p+b_{2}p^{2}&&{\bmod {p}}^{3}\end{aligned}}}$

This motivates the definition of Witt vectors.

Fix a prime number p. A Witt vector^[5] over a commutative ring ${\displaystyle R}$ (relative to the prime ${\displaystyle p}$) is a sequence ${\displaystyle (X_{0},X_{1},X_{2},\ldots )}$ of elements of ${\displaystyle R}$. Define the Witt polynomials ${\displaystyle W_{i}}$ by

1. ${\displaystyle W_{0}=X_{0}}$
2. ${\displaystyle W_{1}=X_{0}^{p}+pX_{1}}$
3. ${\displaystyle W_{2}=X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}}$

and in general

${\displaystyle W_{n}=\sum _{i=0}^{n}p^{i}X_{i}^{p^{n-i}}.}$

The ${\displaystyle W_{n}}$ are called the ghost components of the Witt vector ${\displaystyle (X_{0},X_{1},X_{2},\ldots )}$, and are usually denoted by ${\displaystyle X^{(n)}}$; taken together, the ${\displaystyle W_{n}}$ define the ghost map to ${\textstyle \prod _{i=0}^{\infty }R}$. If ${\textstyle R}$ is p-torsionfree, then the ghost map is injective and the ghost components can be thought of as an alternative coordinate system for the ${\displaystyle R}$-module of sequences (though note that the ghost map is not surjective unless ${\textstyle R}$ is p-divisible).

The ring of (p-typical) Witt vectors ${\displaystyle W(R)}$ is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring ${\displaystyle R}$ into a ring such that:

1. the sum and product are given by polynomials with integer coefficients that do not depend on ${\displaystyle R}$, and
2. projection to each ghost component is a ring homomorphism from the Witt vectors over ${\displaystyle R}$, to ${\displaystyle R}$.

In other words,

• ${\displaystyle (X+Y)_{i}}$ and ${\displaystyle (XY)_{i}}$ are given by polynomials with integer coefficients that do not depend on R, and

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

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

${\displaystyle (X_{0},X_{1},\ldots )+(Y_{0},Y_{1},\ldots )=(X_{0}+Y_{0},X_{1}+Y_{1}-((X_{0}+Y_{0})^{p}-X_{0}^{p}-Y_{0}^{p})/p,\ldots )}$

${\displaystyle (X_{0},X_{1},\ldots )\times (Y_{0},Y_{1},\ldots )=(X_{0}Y_{0},X_{0}^{p}Y_{1}+X_{1}Y_{0}^{p}+pX_{1}Y_{1},\ldots )}$

These are to be understood as shortcuts for the actual formulas: if for example the ring ${\displaystyle R}$ has characteristic ${\displaystyle p}$, the division by ${\displaystyle p}$ in the first formula above, the one by ${\displaystyle p^{2}}$ that would appear in the next component and so forth, do not make sense. However, if the ${\displaystyle p}$-power of the sum is developed, the terms ${\displaystyle X_{0}^{p}+Y_{0}^{p}}$ are cancelled with the previous ones and the remaining ones are simplified by ${\displaystyle p}$, no division by ${\displaystyle p}$ remains and the formula makes sense. The same consideration applies to the ensuing components.

As would be expected, the identity element in the ring of Witt vectors ${\displaystyle W(A)}$ is the element

${\displaystyle {\underline {1}}=(1,0,0,\ldots )}$

Adding this element to itself gives a non-trivial sequence, for example in ${\displaystyle W(\mathbb {F} _{5})}$,

${\displaystyle {\underline {1}}+{\underline {1}}=(2,4,\ldots )}$

since

${\displaystyle {\begin{aligned}2&=1+1\\4&=-{\frac {32-1-1}{5}}\mod 5\\&\cdots \end{aligned}}}$

which is not the expected behavior, since it doesn't equal ${\displaystyle {\underline {2}}}$. But, when we reduce with the map ${\displaystyle m:W(\mathbb {F} _{5})\to \mathbb {F} _{5}}$, we get ${\displaystyle m(\omega (1)+\omega (1))=m(\omega (2))}$. Note if we have an element ${\displaystyle x\in A}$ and an element ${\displaystyle a\in W(A)}$ then

${\displaystyle {\underline {x}}a=(xa_{0},x^{p}a_{1},\ldots ,x^{p^{n}}a_{n},\ldots )}$

showing multiplication also behaves in a highly non-trivial manner.

• The Witt ring of any commutative ring ${\displaystyle R}$ in which ${\displaystyle p}$ is invertible is just isomorphic to ${\displaystyle R^{\mathbb {N} }}$ (the product of a countable number of copies of ${\displaystyle R}$). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to ${\displaystyle R^{\mathbb {N} }}$, and if ${\displaystyle p}$ is invertible this homomorphism is an isomorphism.

• The Witt ring ${\displaystyle W(\mathbb {F} _{p})\cong \mathbb {Z} _{p}}$ of the finite field of order ${\displaystyle p}$ is the ring of ${\displaystyle p}$-adic integers written in terms of the Teichmüller representatives, as demonstrated above.

• The Witt ring ${\displaystyle W(\mathbb {F} _{q})\cong {\mathcal {O}}_{K}}$ of a finite field of order ${\displaystyle p^{n}}$ is the ring of integers of the unique unramified extension of degree ${\displaystyle n}$ of the ring of ${\displaystyle p}$-adic numbers ${\displaystyle K/\mathbb {Q} _{p}}$. Note ${\displaystyle K\cong \mathbb {Q} _{p}(\mu _{q-1})}$ for ${\displaystyle \mu _{q-1}}$ the ${\displaystyle (q-1)}$-st root of unity, hence ${\displaystyle W(\mathbb {F} _{q})\cong \mathbb {Z} _{p}[\mu _{q-1}]}$.

The Witt polynomials for different primes ${\displaystyle p}$ are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime ${\displaystyle p}$). Define the universal Witt polynomials ${\displaystyle W_{n}}$ for ${\displaystyle n\geq 1}$ by

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

and in general

${\displaystyle W_{n}=\sum _{d|n}dX_{d}^{n/d}.}$

Again, ${\displaystyle (W_{1},W_{2},W_{3},\ldots )}$ is called the vector of ghost components of the Witt vector ${\displaystyle (X_{1},X_{2},X_{3},\ldots )}$, and is usually denoted by ${\displaystyle (X^{(1)},X^{(2)},X^{(3)},\ldots )}$.

We can use these polynomials to define the ring of universal Witt vectors or big Witt ring of any commutative ring ${\displaystyle R}$ in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring ${\displaystyle R}$).

Witt also provided another approach using generating functions.^[6]

Let ${\displaystyle X}$ be a Witt vector and define

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

For ${\displaystyle n\geq 1}$ let ${\displaystyle {\mathcal {I}}_{n}}$ denote the collection of subsets of ${\displaystyle \{1,2,\ldots ,n\}}$ whose elements add up to ${\displaystyle n}$. Then

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

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

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

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

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

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

${\displaystyle Z_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i}-\sum _{I\in {\mathcal {I}}_{n},I\neq \{n\}}(-1)^{|I|}\prod _{i\in I}{Z_{i}}.}$

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

If we set ${\displaystyle W=XY}$ then

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

But

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

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

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

So that

${\displaystyle f_{W}(t)=\prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}=\sum _{n\geq 0}D_{n}t^{n},}$

where ${\displaystyle D_{n}}$ are polynomials of ${\displaystyle X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.}$ So by similar induction, suppose

${\displaystyle f_{W}(t)=\prod _{n\geq 1}(1-W_{n}t^{n}),}$

then ${\displaystyle W_{n}}$ can be solved as polynomials of ${\displaystyle X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.}$

The map taking a commutative ring ${\displaystyle R}$ to the ring of Witt vectors over ${\displaystyle R}$ (for a fixed prime ${\displaystyle p}$) 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 ${\displaystyle \operatorname {Spec} (\mathbb {Z} ).}$ 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 ${\displaystyle R}$ to the set ${\displaystyle R^{n}}$ is represented by the affine space ${\displaystyle \mathbb {A} _{\mathbb {Z} }^{n}}$, and the ring structure on ${\displaystyle R^{n}}$ makes ${\displaystyle \mathbb {A} _{\mathbb {Z} }^{n}}$ into a ring scheme denoted ${\displaystyle {\underline {\mathcal {O}}}^{n}}$. From the construction of truncated Witt vectors, it follows that their associated ring scheme ${\displaystyle \mathbb {W} _{n}}$ is the scheme ${\displaystyle \mathbb {A} _{\mathbb {Z} }^{n}}$ with the unique ring structure such that the morphism ${\displaystyle \mathbb {W} _{n}\to {\underline {\mathcal {O}}}^{n}}$ given by the Witt polynomials is a morphism of ring schemes.

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 ${\displaystyle G_{a}}$. The analogue of this for fields of characteristic ${\displaystyle p}$ 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 ${\displaystyle p}$, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

André Joyal explicated the universal property of the (p-typical) Witt vectors.^[7] The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic p ring to characteristic 0 together with a lift of its Frobenius endomorphism.^[8] To make this precise, define a ${\displaystyle \delta }$-ring ${\textstyle (R,\delta )}$ to consist of a commutative ring ${\textstyle R}$ together with a map of sets ${\textstyle \delta :R\to R}$ that is a p-derivation, so that ${\textstyle \delta }$ satisfies the relations

• ${\displaystyle \delta (0)=\delta (1)=0}$;
• ${\displaystyle \delta (xy)=x^{p}\delta (y)+y^{p}\delta (x)+p\delta (x)\delta (y)}$;
• ${\displaystyle \delta (x+y)=\delta (x)+\delta (y)+{\frac {x^{p}+y^{p}-(x+y)^{p}}{p}}}$.

The definition is such that given a ${\displaystyle \delta }$-ring ${\textstyle (R,\delta )}$, if one defines the map ${\textstyle \phi :R\to R}$ by the formula ${\textstyle \phi (x)=x^{p}+p\delta (x)}$, then ${\textstyle \phi }$ is a ring homomorphism lifting Frobenius on ${\displaystyle R/p}$. Conversely, if ${\textstyle R}$ is p-torsionfree, then this formula uniquely defines the structure of a ${\displaystyle \delta }$-ring on ${\textstyle R}$ from that of a Frobenius lift. One may thus regard the notion of ${\displaystyle \delta }$-ring as a suitable replacement for a Frobenius lift in the non-p-torsionfree case.

The collection of ${\displaystyle \delta }$-rings and ring homomorphisms thereof respecting the ${\displaystyle \delta }$-structure assembles to a category ${\textstyle \mathrm {CRing} _{\delta }}$. One then has a forgetful functor$${\displaystyle U:\mathrm {CRing} _{\delta }\to \mathrm {CRing} }$$whose right adjoint identifies with the functor ${\textstyle W}$ of Witt vectors. In fact, the functor ${\textstyle U}$ creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor; from this, it is not hard to show that ${\textstyle \mathrm {CRing} _{\delta }}$ inherits local presentability from ${\displaystyle \mathrm {CRing} }$ so that one can construct the functor ${\textstyle W}$ by appealing to the adjoint functor theorem.

One further has that ${\textstyle W}$ restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its essential image then consists of those ${\displaystyle \delta }$-rings that are perfect (in the sense that the associated map ${\textstyle \phi }$ is an isomorphism) and whose underlying ring is p-adically complete.^[9]

• p-derivation
• Formal group
• Artin–Hasse exponential
• Necklace ring

• Notes on Witt vectors: a motivated approach - Basic notes giving the main ideas and intuition. Best to start here!
• The Theory of Witt Vectors - Elementary introduction to the theory.
• Complexe de de Rham-Witt et cohomologie cristalline - Note he uses a different but equivalent convention as in this article. Also, the main points in the introduction are still valid.

• Mumford, David (1966-08-21), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237, section II.6
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1035-1, ISBN 978-0-387-96648-9, MR 0918564
• Greenberg, Marvin J. (1969). Lectures on Forms in Many Variables. New York and Amsterdam: Benjamin. ASIN B0006BX17M. MR 0241358.

• Dolgachev, Igor V. (2001) [1994], "Witt vector", Encyclopedia of Mathematics, EMS Press
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
• 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 die Reine und Angewandte Mathematik (in German), 1937 (176): 126–140, doi:10.1515/crll.1937.176.126