# nLab ring of Witt vectors

This article is about of groups of Witt vectors and rings of Witt vectors; which are often called just Witt groups and Witt rings. However, there is also a different notion of Witt group? and Witt ring.

# Contents

## Idea

Rings of Witt vectors are the co-free Lambda-rings.

In components, a Witt vector is an infinite sequence of elements of a given commutative ring $k$. There is a ring structure on the set $W\left(k\right)$ of Witt vectors of $k$ and $W\left(k\right)$ is therefore called the Witt ring of $k$. The multiplication is defined by means of Witt polynomials ${w}_{i}$ for every natural number $i$. If the characteristic of $k$ is $0$ the Witt ring of $k$ is sometimes called universal Witt ring to distinguish it from the case where $k$ is of prime characteristic and a similar but different construction is of interest.

A p-adic Witt vector is an infinite sequence of elements af a commutative ring of prime characteristic $p$. There exists a ring structure whose construction parallels that in characteristic $0$ except that only Witt polynomials ${w}_{{p}^{l}}$ whose index is a power of $p$ are taken.

More abstractly, the ring of Witt vectors carries the structure of a Lambda-ring and the construction $W:k↦W\left(k\right)$ of the Witt ring $W\left(k\right)$ on a commutative ring $k$ is right adjoint to the forgetful functor from Lambda-rings to commutative rings. Hence rings of Witt vectors are the co-free Lambda-rings.

Moreover $W\left(-\right)$ is representable by Symm, the ring of symmetric functions which is a Hopf algebra and consequently $W$ is a group scheme. This is explained at Lambda-ring.

The construction of Witt vectors gives a functorial way to lift a commutative ring $A$ of prime characteristic $p$ to a commutative ring $W\left(A\right)$ of characteristic 0. Since this construction is functorial, it can be applied to the structure sheaf of an algebraic variety. In interesting special cases the resulting ring $W\left(A\right)$ has even more desirable properties: If $A$ is a perfect field then $W\left(A\right)$ is a discrete valuation?. This is partly due to the fact that the construction of $W\left(A\right)$ involves a ring of power series and a ring of power series over a field is always a discrete valuation ring.

There is a generalization to non-commutative Witt vectors, however these only carry a group- but no ring structure.

The Lubin-Tate ring in Lubin-Tate theory is a polynomial ring on a ring of Witt vectors and this way Witt vectors control much of chromatic homotopy theory.

## Motivation

In an expansion of a $p$-adic number $a=\Sigma {a}_{i}{p}^{i}$ the ${a}^{i}$ are called digits. Usually these digits are defined to be taken elements of the set $\left\{0,1,\dots ,p-1\right\}$.

Equivalently the digits can be defined to be taken from the set ${T}_{p}:=\left\{x\mid {x}^{p-1}=1\right\}\cup \left\{0\right\}$. Elements from this set are called Teichmüller digits or Teichmüller representatives.

The set $T$ is in bijection with the finite field ${F}_{p}$. The set $W\left({F}_{p}\right)$ of (countably) infinite sequences of elements in ${F}_{p}$ hence is in bijection to the set ${ℤ}_{p}$ of $p$-adic integers. There is a ring structure on $W\left({F}_{p}\right)$ called Witt ring structure such that all ”truncated expansion polynomials” ${\Phi }_{n}={X}^{{p}^{n}}+{\mathrm{pX}}^{{p}^{n-1}}+{p}^{2}{X}^{{p}^{n-2}}+\dots +{p}^{n}X$ called Witt polynomials are morphisms

${\Phi }_{n}:W\left({F}_{p}\right)\to {ℤ}_{p}$\Phi_n:W(F_p)\to \mathbb{Z}_p

of groups.

## Definition

We first give the

and then discuss the

### In components

#### The ring structure

###### Definition

Let $k$ be a commutative ring.

If the characteristic of $k$ is $0$ then the Witt ring $W\left(k\right)$ of $k$ is defined defined by the addition

$\left({a}_{1},{a}_{2},\dots ,\right)+\left({b}_{1},{b}_{2},\dots \right):=\left({\Sigma }_{1}\left({a}_{1},{b}_{1}\right),{\Sigma }_{2}\left({a}_{1},{a}_{2},{b}_{1},{b}_{2}\right),\dots \right)$(a_1, a_2, \ldots, )+(b_1, b_2, \ldots) :=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)

and the multiplication

$\left({a}_{1},{a}_{2},\dots \right)\cdot \left({b}_{1},{b}_{2},\dots \right):=\left({\Pi }_{1}\left({a}_{1},{b}_{1}\right),{\Pi }_{2}\left({a}_{1},{a}_{2},{b}_{1},{b}_{2}\right),\dots \right)$(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots ) :=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )

If $k$ is of prime characteristic $p$ we index the defining formulas by ${p}^{1},{p}^{2},{p}^{3},\dots$ instead of $1,2,3,\dots$.

Here the ${\Sigma }_{i}$ are called addition polynomials and the ${\Pi }_{i}$ are called multiplication polynomials, these are described below.

#### The Witt polynomials, the ${\Sigma }_{i}$, the ${\Pi }_{i}$, phantom components

let ${x}_{1},{x}_{2},\dots$ be a collection of variables. We can define an infinite collection of polynomials in $ℤ\left[{x}_{1},{x}_{2},\dots \right]$ using the following formulas:

${w}_{1}\left(X\right)={x}_{1}$

${w}_{2}\left(X\right)={x}_{1}^{2}+2{x}_{2}$

${w}_{3}\left(X\right)={x}_{1}^{3}+3{x}_{3}$

${w}_{4}\left(X\right)={x}_{1}^{4}+2{x}_{2}^{2}+4{x}_{4}$

and in general ${w}_{n}\left(X\right)=\sum _{d\mid n}{\mathrm{dx}}_{n}^{n/d}$. The value ${w}_{n}\left(w\right)$ of the $n$-th Witt polynomial in some element $w\in W\left(k\right)$ of the Witt ring of $k$ is sometimes called the $n$-th phantom component of $w$ or the $n$-th ghost component of $w$.

Now let $\varphi \left({z}_{1},{z}_{2}\right)\in ℤ\left[{z}_{1},{z}_{2}\right]$. This just an arbitrary two variable polynomial with coefficients in $ℤ$.

We can define new polynomials ${\Phi }_{i}\left({x}_{1},\dots {x}_{i},{y}_{1},\dots {y}_{i}\right)$ such that the following condition is met $\varphi \left({w}_{n}\left({x}_{1},\dots ,{x}_{n}\right),{w}_{n}\left({y}_{1},\dots ,{y}_{n}\right)\right)={w}_{n}\left({\Phi }_{1}\left({x}_{1},{y}_{1}\right),\dots ,{\Phi }_{n}\left({x}_{1},\dots {x}_{n},{y}_{1},\dots ,{y}_{n}\right)\right)$.

In short we’ll notate this $\varphi \left({w}_{n}\left(X\right),{w}_{n}\left(Y\right)\right)={w}_{n}\left(\Phi \left(X,Y\right)\right)$. The first thing we need to do is make sure that such polynomials exist. Now it isn’t hard to check that the ${x}_{i}$ can be written as a $ℚ$-linear combination of the ${w}_{n}$ just by some linear algebra.

${x}_{1}={w}_{1}$, and ${x}_{2}=\frac{1}{2}{w}_{2}+\frac{1}{2}{w}_{1}^{2}$, etc. so we can plug these in to get the existence of such polynomials with coefficients in $ℚ$. It is a fairly tedious lemma to prove that the coefficients ${\Phi }_{i}$ are actually in $ℤ$, so we won’t detract from the construction right now to prove it.

#### The addition- and multiplication polynomials

Define yet another set of polynomials ${\Sigma }_{i}$, ${\Pi }_{i}$ and ${\iota }_{i}$ by the following properties:

${w}_{n}\left(\Sigma \right)={w}_{n}\left(X\right)+{w}_{n}\left(Y\right)$, ${w}_{n}\left(\Pi \right)={w}_{n}\left(X\right){w}_{n}\left(Y\right)$ and ${w}_{n}\left(\iota \right)=-{w}_{n}\left(X\right)$.

We now can construct $W\left(A\right)$, the ring of generalized Witt vectors over $A$. Define $W\left(A\right)$ to be the set of all infinite sequences $\left({a}_{1},{a}_{2},\dots \right)$ with entries in $A$. Then we define addition and multiplication by $\left({a}_{1},{a}_{2},\dots ,\right)+\left({b}_{1},{b}_{2},\dots \right)=\left({\Sigma }_{1}\left({a}_{1},{b}_{1}\right),{\Sigma }_{2}\left({a}_{1},{a}_{2},{b}_{1},{b}_{2}\right),\dots \right)$ and $\left({a}_{1},{a}_{2},\dots \right)\cdot \left({b}_{1},{b}_{2},\dots \right)=\left({\Pi }_{1}\left({a}_{1},{b}_{1}\right),{\Pi }_{2}\left({a}_{1},{a}_{2},{b}_{1},{b}_{2}\right),\dots \right)$.

### Universal characterization

###### Theorem

The assignment

$W\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}k↦W\left(k\right)$W \;\colon\; k\mapsto W(k)

is a functor

$W\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\mathrm{CRing}⟶\Lambda \mathrm{Ring}$W \;\colon\; CRing \longrightarrow \Lambda Ring

from the category of commutative rings to that of Lambda-rings.

Composed with the forgetful functor

$U\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\Lambda \mathrm{Ring}⟶\mathrm{CRing}$U \;\colon\; \Lambda Ring \longrightarrow CRing

this is the unique endofunctor $W\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\mathrm{CRing}⟶\mathrm{CRing}$ such that all Witt polynomials

${w}_{n}:\left\{\begin{array}{l}W\left(A\right)\to A\\ a↦{w}_{n}\left(a\right)\end{array}$w_n : \begin{cases} W(A)\to A \\ a\mapsto w_n(a) \end{cases}

are homomorphisms of rings.

###### Proof

There is a nice trick to prove that $W\left(A\right)$ is a ring when $A$ is a $ℚ$-algebra. Just define $\psi :W\left(A\right)\to {A}^{ℕ}$ by $\left({a}_{1},{a}_{2},\dots \right)↦\left({w}_{1}\left(a\right),{w}_{2}\left(a\right),\dots \right)$. This is a bijection and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know $W\left(A\right)$ is a ring. Also, $w\left(0,0,\dots \right)=\left(0,0,\dots \right)$, so $\left(0,0,\dots \right)$ is the additive identity, $W\left(1,0,0,\dots \right)=\left(1,1,1,\dots \right)$ which shows $\left(1,0,0,\dots \right)$ is the multiplicative identity, and $w\left({\iota }_{1}\left(a\right),{\iota }_{2}\left(a\right),\dots \right)=\left(-{a}_{1},-{a}_{2},\dots \right)$, so we see $\left({\iota }_{1}\left(a\right),{\iota }_{2}\left(a\right),\dots \right)$ is the additive inverse.

We can actually get this idea to work for any characteristic $0$ ring by considering the embedding $A\to A\otimes ℚ$. We have an induced injective map $W\left(A\right)\to W\left(A\otimes ℚ\right)$. The addition and multiplication is defined by polynomials over $ℤ$, so these operations are preserved upon tensoring with $ℚ$. We just proved above that $W\left(A\otimes ℚ\right)$ is a ring, so since $\left(0,0,\dots \right)↦\left(0,0,\dots \right)$ and $\left(1,0,0,\dots \right)↦\left(1,0,0,\dots \right)$ and the map preserves inverses we get that the image of the embedding $W\left(A\right)\to W\left(A\otimes ℚ\right)$ is a subring and hence $W\left(A\right)$ is a ring.

Lastly, we need to prove this for positive characteristic rings. Choose a characteristic $0$ ring that surjects onto $A$, say $B\to A$. Then since the induced map again preserves everything and $W\left(B\right)\to W\left(A\right)$ is surjective, the image is a ring and hence $W\left(A\right)$ is a ring.

###### Proposition

The construction of the ring of Witt vectors $W\left(k\right)$ on a given commutative ring $k$ is the right adjoint to the forgetful functor $U$ from Lambda-rings to commutative rings

$\left(U⊣W\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\mathrm{CRing}\stackrel{\stackrel{U}{←}}{\underset{W}{⟶}}\Lambda \mathrm{Ring}\phantom{\rule{thinmathspace}{0ex}}.$(U \dashv W) \;\colon\; CRing \stackrel{\overset{U}{\leftarrow}}{\underset{W}{\longrightarrow}} \Lambda Ring \,.

Hence rings of Witt-vectors are the co-free Lambda-rings.

This statement appears in (Hazewinkel 08, p. 87, p. 97).

###### Remark

On the other hand, the free Lambda-ring (on one generator) (hence the left adjoint construction to the forgetful functor) is the ring of symmetric functions.

This statement appears in (Hazewinkel 08, p. 98).

## Properties

### Operations on the p-adic Witt vectors

On untruncated $p$-adic Witt vectors there are two operations, the Frobenius morphism and the Verschiebung morphism? satisfying relations (Lemma 1) being constitutive for the definition of the Dieudonné ring: In fact the Dieudonné ring is generated by two objects satisfying these relations.

Also the $n$-truncations of a Witt ring are rings since by definition the ring operations (addition and multiplication) of the first $n$ components only involve the first $n$ components. We have $W\simeq {\mathrm{lim}}_{n}{W}_{n}$ and the projection map $W\left(A\right)\to {W}_{n}\left(A\right)$ is a ring homomorphism. We also have operations on the truncated Witt rings.

#### The shift map

For $W\left(k\right)$ as for every $k$-scheme we have the Verschiebung morphism?. It is defined to be the adjoint operation to the Frobenius morphism. For $W\left(k\right)$ the Verschiebung morphism coincides with the shift $\left({a}_{0},{a}_{1},\dots \right)↦\left(0,{a}_{0},{a}_{1},\dots \right)$

For the truncated Witt rings and the shift operation $V:{W}_{n}\left(k\right)\to {W}_{n+1}\left(k\right)$ the Verschiebung morphism equals the $\mathrm{VR}=\mathrm{RV}$ where $R$ is the restriction map.

#### The restriction map

The restriction map $R:{W}_{n+1}\left(A\right)\to {W}_{n}\left(A\right)$ is given by $\left({a}_{0},\dots ,{a}_{n}\right)↦\left({a}_{0},\dots ,{a}_{n-1}\right)$.

#### The Frobenius morphism

The Frobenius endomorphism $F:{W}_{n}\left(A\right)\to {W}_{n}\left(A\right)$ is given by $\left({a}_{0},\dots ,{a}_{n-1}\right)↦\left({a}_{0}^{p},\dots ,{a}_{n-1}^{p}\right)$. This is also a ring map, but only because of our necessary assumption that $A$ is of characteristic $p$.

Just by brute force checking on elements we see a few relations between these operations, namely that $V\left(x\right)y=V\left(xF\left(R\left(y\right)\right)\right)$ and $\mathrm{RVF}=\mathrm{FRV}=\mathrm{RFV}=p$ the multiplication by $p$ map.

### Duality of finite Witt groups

For a $k$-ring $R$ let ${W}^{\prime }\left(R\right)$ denote the ideal in $W\left(R\right)$ consisting of sequences $x=\left({x}_{n}{\right)}_{n}$ of nilpotent elements in $W\left(R\right)$ such that ${x}_{n}=0$ for large $n$.

Let $E$ denote the Artin-Hasse exponential?. Then we have $E\left(x,1\right)$ is a polynomial for $x\in {W}^{\prime }\left(R\right)$ and

$E\left(-,1\right):\left\{\begin{array}{l}{W}^{\prime }\to {\mu }_{k}\\ w↦E\left(w,1\right)\end{array}$E(-,1):\begin{cases}W^\prime\to \mu_k \\ w\mapsto E(w,1)\end{cases}

is a morphism of group schemes to the multiplicative group scheme ${\mu }_{k}$.

###### Proposition

a) ${W}^{\prime }\left(R\right)$ is an ideal in $W\left(R\right)$.

b) $E\left(-,1\right):{W}^{\prime }\to {\mu }_{k}$ is an morphism of group schemes.

c) The morphism

$\left\{\begin{array}{l}W×{W}^{\prime }\to {\mu }_{k}\\ \left(x,y\right)↦E\left(\mathrm{xy},1\right)\end{array}$\begin{cases} W\times W^\prime\to \mu_k \\ (x,y)\mapsto E(xy,1) \end{cases}

is bilinear and gives an isomorphism of group schemes

${W}^{\prime }\to D\left(W\right)$W^\prime\to D(W)

where $D$ denotes the Cartier dual of $W$ (maybe it is equivalently the Pontryagin dual of the underlying group of the (plain) ring $W$). That this map is a morphism of group schemes follows from the definition of the Cartier dual.

d) For $x\in W\left(R\right)$ and $y\in {W}^{\prime }\left(R\right)$ we have $E\left(\mathrm{xy},1\right)\in {R}^{×}$ and

$E\left({V}^{n}xy,1\right)=E\left({T}^{n}\left(x{F}^{n}y\right),1\right)=E\left(x{F}^{n}y,1\right)$E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1)
###### Definition and Theorem

Let $\mathrm{ker}\left({F}_{n}^{m}\right):=\mathrm{ker}\left({F}^{m}:{W}_{\mathrm{nk}}\to {W}_{\mathrm{nk}}\right)$ denote the kernel of $m$-times iterated Frobenius endomorphism of the $n$-truncated Witt ring.Let

${\sigma }_{n}:\left\{\begin{array}{l}{W}_{\mathrm{nk}}\to {W}_{k}\\ \left({\alpha }_{0},\dots ,{\alpha }_{n-1}\right)↦\left({\alpha }_{0},\dots ,{\alpha }_{n-1},0,\dots \right)\end{array}$\sigma_n:\begin{cases} W_{nk}\to W_k \\ (\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots) \end{cases}

be the section of the restriction ${R}_{n}:{W}_{k}\to {W}_{\mathrm{nk}}$. ${\sigma }_{n}$ sends $\mathrm{ker}\left({F}_{n}^{m}\right)$ in ${W}^{\prime }$. Note that ${\sigma }_{n}$ is not a morphism of groups.

We define the bilinear map

$<-,->:\left\{\begin{array}{l}\mathrm{ker}\left({F}_{n}^{m}\right)×\mathrm{ker}\left({F}_{m}^{n}\right)\to {R}^{×}\\ ↦E\left({\sigma }_{n}\left(x\right){\sigma }_{m}\left(y\right),1\right)\end{array}$\lt-,-\gt:\begin{cases} ker(F^m_n)\times ker(F^n_m)\to R^\times \\ \lt x,y\gt\mapsto E(\sigma_n(x)\sigma_m(y),1) \end{cases}

then $$ is bilinear and gives an isomorphism

$\mathrm{ker}\left({F}_{n}^{m}\right)\simeq D\left(\mathrm{ker}\left({F}_{m}^{n}\right)$ker(F^m_n)\simeq D(ker(F^n_m)

and satisfies

$=$\lt x,t y\gt=\lt f x,y\gt
$=$\lt x,r y\gt=\lt i x,y\gt

where the morphisms are

$\begin{array}{ccc}\mathrm{ker}\left({F}_{n}^{m}\right)& \stackrel{t}{\to }& \mathrm{ker}\left({F}_{n+1}^{m}\right)\\ {↓}^{f}& & {↓}^{r}\\ \mathrm{ker}\left({F}_{n}^{m-1}\right)& \stackrel{i}{↪}& \mathrm{ker}\left({F}_{n}^{m}\right)\end{array}$\array{ ker(F^m_n) &\stackrel{t}{\to}& ker(F^m_{n+1}) \\ \downarrow^f&&\downarrow^r \\ ker(F^{m-1}_n) &\stackrel{i}{\hookrightarrow}& ker(F^m_n) }

where $i$ is the canonical inclusion, and $r,f,t$ are induced by $R,F,T$ where $F$ is Frobenius, $T$ is Verschiebung and $R:W\to {W}_{n}$ is restriction. $i$ and $t$ are monomorphisms, $f$ and $r$ are epimorphisms, and for $\mathrm{ker}\left({F}_{m}^{n}\right)$ we have $F=\mathrm{if}$ and $V=\mathrm{rt}$.

References: Pink, §25, Demazure, III.4

## Properties of the Witt group

The group of universal (i.e. not $p$-adic) Witt vectors equals $W\left(k\right)=1+k\left[\left[X\right]\right]$ i.e. the multiplicative group of power series in one variable $X$ with constant term $1$.

## Properties of the Witt ring

###### Theorem

Let $k$ be a perfect field of prime characteristic $p$.

Then $W\left(k\right)$ is a discrete valuation ring with maximal ideal generated by $p$. From the above we see that $\mathrm{pW}\left(k\right)=\left(0,{a}_{0}^{p},{a}_{1}^{p},\dots \right)$. This clearly gives $W\left(k\right)/\mathrm{pW}\left(k\right)\simeq k$.

Also, $W\left(k\right)/{p}^{\mathrm{nW}}\left(k\right)\simeq {W}_{n}\left(k\right)$. Thus the completion of $W\left(k\right)$ with respect to the maximal ideal is just $\mathrm{lim}{W}_{n}\left(k\right)\simeq W\left(k\right)$ which shows that $W\left(k\right)$ is a complete discrete valuation ring.

## Examples

### Basic examples

• ${W}_{{p}^{\infty }}\left({𝔽}_{p}\right)\simeq {ℤ}_{p}$ the $p$-adic integers.

• ${W}_{{p}^{\infty }}\left({𝔽}_{{p}^{n}}\right)$ is the unique unramified extension? of ${ℤ}_{p}$ of degree $n$.

### Lubin-Tate ring

The Lubin-Tate ring in Lubin-Tate theory is a power series ring over a Witt ring and this way Witt rings govern much of chromatic homotopy theory.

As an Abelian group $W\left(A\right)$ is isomorphic to the group of curves in the one-dimensional multiplicative formal group. In this way there is a Witt-vector-like Abelian-group-valued functor associated to every one-dimensional formal group. For special cases, such as the Lubin–Tate formal groups, this gives rise to ring-valued functors called ramified Witt vectors. (eom)

## References

### Original texts and classical surveys

witt vectors were introduced in

• Ernst Witt, Zyklische Körper und Algebren der Characteristik $p$ vom Grad ${p}^{n}$. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik ${p}^{n}$, J. Reine Angew. Math. , 176 (1936) pp. 126–140, (web)

In the context of formal group laws they were used in

• Jean Dieudonné, Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique $p>0$ VII” Math. Ann. , 134 (1957) pp. 114–133