Ring of Witt vectors

Definition

The term ring of Witt vectors or Witt ring is a commutative unital ring defined in the context of a prime number $p$ and a commutative unital ring $R$. There are two version of this:

• The ring of Witt vectors, by default, refers to infinite sequences, i.e., Witt vectors of infinite length (the detailed definition is given below).
• The ring of Witt vectors of length $d$ refers to the quotient of this ring obtained by simply looking at the first $d$ coordinates of the vector. The operations are defined in a manner that the first $d$ coordinates of the sum and product depend only on the first $d$ coordinates of the vectors being added or multiplied. Thus, it is legitimate to pass to quotients in this manner.

There is a related notion called the ring of universal Witt vectors that makes sense in the context of any commutative unital ring $R$ and does not require specification of any prime number.

Definition of the ring of Witt vectors of infinite length

We first define the Witt polynomials as the following polynomials in $\mathbb {Z} [X_{0},X_{1},X_{2},\dots ,]$ (and hence interpretable in $R[X_{0},X_{1},X_{2},\dots ,]$):

$W_{n}(X):=\sum _{i=0}^{n}p^{i}X_{i}^{p^{n-i}}$

The first few Witt polynomials are given below:

$n$	$W_{n}$
0	$X_{0}$
1	$X_{0}^{p}+pX_{1}$
2	$X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}$
3	$X_{0}^{p^{3}}+pX_{1}^{p^{2}}+p^{2}X_{2}^{p}+p^{3}X_{3}$

We now consider the set $(X_{0},X_{1},X_{2},\dots ,X_{n},\dots ),X_{i}\in R$ of all sequences over $R$. We call the elements of this set Witt vectors, and we define the addition and multiplication as follows:

Addition and multiplication are defined in the unique manner so as to make the set of Witt vectors a commutative unital ring with the following two properties: (a) each Witt polynomial is a ring homomorphism from that ring to $R$, and (b) both addition and multiplication are given by polynomials with integer coefficients that depend on $p$ but not on $R$.

Explicitly, the sum and product of the Witt vectors $(X_{0},X_{1},X_{2},\dots ,)$ and $(Y_{0},Y_{1},Y_{2},\dots ,)$ are given by the following formulas:

$n$	$n^{th}$ coordinate of sum	$n^{th}$ coordinate of product
0	$X_{0}+Y_{0}$	$X_{0}Y_{0}$
1	$X_{1}+Y_{1}-\sum _{i=1}^{p-1}{\frac {(p-1)!}{i!(p-i)!}}X_{0}^{i}Y_{0}^{p-i}$, which can be thought of as $X_{1}+Y_{1}+{\frac {(X_{0}^{p}+Y_{0}^{p})-(X_{0}+Y_{0})^{p}}{p}}$ (although the latter notation does not directly make sense unless $p$ is invertible in $R$).	$X_{0}^{p}Y_{1}+Y_{0}^{p}X_{1}+pX_{1}Y_{1}$

Some elaboration on why these choices work is provided below:

$n$	$W_{n}(X)$	$W_{n}(Y)$	$W_{n}(X)+W_{n}(Y)$, which equals $W_{n}$ of the Witt sum	$W_{n}(X)W_{n}(Y)$, which equals $W_{n}$ of the Witt product	Comment
0	$X_{0}$	$Y_{0}$	$X_{0}+Y_{0}$	$X_{0}Y_{0}$	Zeroth Witt polynomial is projection onto zeroth coordinate; pointwise operations mean this is a homomorphism.
1	$X_{0}^{p}+pX_{1}$	$Y_{0}^{p}+pY_{1}$	$X_{0}^{p}+Y_{0}^{p}+p(X_{1}+Y_{1})$, which is the same as $(X_{0}+Y_{0})^{p}+p\left[X_{1}+Y_{1}+{\frac {(X_{0}^{p}+Y_{0}^{p})-(X_{0}+Y_{0})^{p}}{p}}\right]$	$X_{0}^{p}Y_{0}^{p}+pX_{0}^{p}Y_{1}+pY_{0}^{p}X_{1}+p^{2}X_{1}Y_{1}$, which is the same as $(X_{0}Y_{0})^{p}+p(X_{0}^{p}Y_{1}+Y_{0}^{p}X_{1}+pX_{1}Y_{1})$	Things work out.

Truncations to finite length

We can also consider the ring of Witt vectors of length $d$. This is obtained by taking the ring of Witt vectors and projecting it to the first $d$ coordinates $X_{0},X_{1},X_{2},\dots ,X_{d-1}$. The operations all have the property that the formula of the $i^{th}$ coordinate of the sum and product involves only vectors $X_{j},Y_{j},j\leq i$, so the ring operations descend to the quotient and we get a ring structure on the quotient.

Particular cases

Note that $p$ is the prime with respect to which we are defining the Witt polynomials.

Condition on $R$	Conclusion for ring of Witt vectors (infinite)	Conclusion for truncation to length $d$
$p$ is invertible in $R$	Isomorphic to the external direct product of countably many copies of $R$ (i.e., sequences with pointwise sum and product), i.e., $R^{\omega }$	Isomorphic to $R^{d}$
$R$ is the finite prime field $\mathbb {F} _{p}$	Isomorphic to the ring of p-adic integers, where a Witt vector $(X_{0},X_{1},X_{2},\dots )$ is identified with the $p$-adic integer $\sum _{i=0}^{\infty }X_{i}p^{i}$.	Isomorphic to the ring of integers mod $p^{d}$, i.e., the ring $\mathbb {Z} /p^{d}\mathbb {Z}$.
$R$ is a finite field of order $p^{k}$, $k$ a natural number	Isomorphic to an unramified degree $k$ extension of the ring of p-adic integers. There is only one such extension up to isomorphism.	Isomorphic to the Galois ring, which is an unramified extension of $\mathbb {Z} /p^{d}\mathbb {Z}$ of degree $k$, and can be constructed using any minimal polynomial for $\mathbb {F} _{p^{k}}$ over $\mathbb {F} _{p}$.

Facts

• Linear representation of finite group over finite field of coprime characteristic lifts uniquely to Witt ring
• Additive group of ring of Witt vectors inherits algebraic group structure