# 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. Depending on whether one defines the latter via Frobenius lifts at a single prime number $p$ one speaks of $p$-typical Witt vectors, or of big Witt vectors if all primes are considered at once.

In arithmetic geometry the impact of rings of Witt vectors $W(R)$ of a given ring $R$ is that they are like rings formal power series on $Spec(R)$, such as rings of p-adic numbers. For more on this see at arithmetic jet space and at Borger's absolute geometry.

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 $W(k)$ of Witt vectors of $k$ and $W(k)$ 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 $W \colon k\mapsto W(k)$ of the Witt ring $W(k)$ 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 $W(-)$ 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 $W(A)$ 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 $W(A)$ has even more desirable properties: If $A$ is a perfect field then $W(A)$ is a discrete valuation. This is partly due to the fact that the construction of $W(A)$ 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, $W_G$, to any profinite group, $G$, due to Dress and Siebeneicher (DS88), known as Witt-Burnside functor.

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 $\{0,1,\dots,p-1\}$.

Equivalently the digits can be defined to be taken from the set $T_p:=\{x|x^{p-1}=1\}\cup \{0\}$. 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 $W(F_p)$ of (countably) infinite sequences of elements in $F_p$ hence is in bijection to the set $\mathbb{Z}_p$ of $p$-adic integers. There is a ring structure on $W(F_p)$ called Witt ring structure such that all ‘’truncated expansion polynomials’‘ $\Phi_n=X^{p^n}+pX^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X$ called Witt polynomials are morphisms

$\Phi_n:W(F_p)\to \mathbb{Z}_p$

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 $W(k)$ of $k$ is defined defined by the addition

$(a_1, a_2, \ldots, )+(b_1, b_2, \ldots) :=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)$

and the multiplication

$(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots ) :=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$

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 $x_1, x_2, \ldots$ be a collection of variables. We can define an infinite collection of polynomials in $\mathbb{Z}[x_1, x_2, \ldots ]$ using the following formulas:

$w_1(X)=x_1$

$w_2(X)=x_1^2+2x_2$

$w_3(X)=x_1^3+3x_3$

$w_4(X)=x_1^4+2x_2^2+4x_4$

and in general $\displaystyle w_n(X)=\sum_{d|n} dx_d^{n/d}$. The value $w_n(w)$ of the $n$-th Witt polynomial in some element $w\in W(k)$ 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 $\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]$. This just an arbitrary two variable polynomial with coefficients in $\mathbb{Z}$.

We can define new polynomials $\Phi_i(x_1, \ldots x_i, y_1, \ldots y_i)$ such that the following condition is met $\phi(w_n(x_1, \ldots ,x_n), w_n(y_1, \ldots , y_n))=w_n(\Phi_1(x_1, y_1), \ldots , \Phi_n(x_1, \ldots x_n, y_1, \ldots , y_n))$.

In short we’ll notate this $\phi(w_n(X),w_n(Y))=w_n(\Phi(X,Y))$. 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 $\mathbb{Q}$-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 $\mathbb{Q}$. It is a fairly tedious lemma to prove that the coefficients $\Phi_i$ are actually in $\mathbb{Z}$, 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:

$w_n(\Sigma)=w_n(X)+w_n(Y)$, $w_n(\Pi)=w_n(X)w_n(Y)$ and $w_n(\iota)=-w_n(X)$.

We now can construct $W(A)$, the ring of generalized Witt vectors over $A$. Define $W(A)$ to be the set of all infinite sequences $(a_1, a_2, \ldots)$ with entries in $A$. Then we define addition and multiplication by $(a_1, a_2, \ldots, )+(b_1, b_2, \ldots)=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)$ and $(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$.

### Universal characterization

###### Theorem

The assignment

$W \;\colon\; k\mapsto W(k)$

is a functor

$W \;\colon\; CRing \longrightarrow \Lambda Ring$

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

Composed with the forgetful functor

$U \;\colon\; \Lambda Ring \longrightarrow CRing$

this is the unique endofunctor $W \;\colon\; CRing \longrightarrow CRing$ such that all Witt polynomials

$w_n : \begin{cases} W(A)\to A \\ a\mapsto w_n(a) \end{cases}$

are homomorphisms of rings.

###### Proof

There is a nice trick to prove that $W(A)$ is a ring when $A$ is a $\mathbb{Q}$-algebra. Just define $\psi: W(A)\to A^\mathbb{N}$ by $(a_1, a_2, \ldots) \mapsto (w_1(a), w_2(a), \ldots)$. 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 $W(A)$ is a ring. Also, $w(0,0,\ldots)=(0,0,\ldots)$, so $(0,0,\ldots)$ is the additive identity, $W(1,0,0,\ldots)=(1,1,1,\ldots)$ which shows $(1,0,0,\ldots)$ is the multiplicative identity, and $w(\iota_1(a), \iota_2(a), \ldots)=(-a_1, -a_2, \ldots)$, so we see $(\iota_1(a), \iota_2(a), \ldots)$ is the additive inverse.

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

###### Proposition

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

$(U \dashv W) \;\colon\; CRing \stackrel{\overset{U}{\leftarrow}}{\underset{W}{\longrightarrow}} \Lambda Ring \,.$

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 $W\simeq lim_n W_n$ and the projection map $W(A)\to W_n(A)$ is a ring homomorphism. We also have operations on the truncated Witt rings.

#### The shift map

For $W(k)$ as for every $k$-scheme we have the Verschiebung morphism. It is defined to be the adjoint operation to the Frobenius morphism. For $W(k)$ the Verschiebung morphism coincides with the shift $(a_0, a_1,\dots)\mapsto (0, a_0, a_1,\dots)$

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

#### The restriction map

The restriction map $R: W_{n+1}(A)\to W_n(A)$ is given by $(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})$.

#### The Frobenius morphism

The Frobenius endomorphism $F: W_n(A)\to W_n(A)$ is given by $(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)$. 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 $V(x)y=V(x F(R(y)))$ and $RVF=FRV=RFV=p$ the multiplication by $p$ map.

### Duality of finite Witt groups

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

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

$E(-,1):\begin{cases}W^\prime\to \mu_k \\ w\mapsto E(w,1)\end{cases}$

is a morphism of group schemes to the multiplicative group scheme $\mu_k$.

###### Proposition

a) $W^\prime (R)$ is an ideal in $W(R)$.

b) $E(-,1): W^\prime\to \mu_k$ is an morphism of group schemes.

c) The morphism

$\begin{cases} W\times W^\prime\to \mu_k \\ (x,y)\mapsto E(xy,1) \end{cases}$

is bilinear and gives an isomorphism of group schemes

$W^\prime\to D(W)$

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 $x\in W(R)$ and $y\in W^\prime (R)$ we have $E(xy,1)\in R^\times$ and

$E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1)$
###### Definition and Theorem

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

$\sigma_n:\begin{cases} W_{nk}\to W_k \\ (\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots) \end{cases}$

be the section of the restriction $R_n:W_k\to W_{nk}$. $\sigma_n$ sends $ker(F_n^m)$ in $W^\prime$. Note that $\sigma_n$ is not a morphism of groups.

We define the bilinear map

$\lt-,-\gt:\begin{cases} ker(F^m_n)\times ker(F^n_m)\to R^\times \\ \lt x,y\gt\mapsto E(\sigma_n(x)\sigma_m(y),1) \end{cases}$

then $\lt x,y\gt$ is bilinear and gives an isomorphism

$ker(F^m_n)\simeq D(ker(F^n_m)$

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

$\array{ ker(F^m_n) &\stackrel{t}{\to}& ker(F^m_{n+1}) \\ \downarrow^f&&\downarrow^r \\ ker(F^{m-1}_n) &\stackrel{i}{\hookrightarrow}& ker(F^m_n) }$

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 $ker(F^n_m)$ we have $F=if$ and $V=rt$.

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

## Properties of the Witt group

The group of universal (i.e. not $p$-adic) Witt vectors equals $W(k)= 1+k [ [ X] ]$ 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 $W(k)$ is a discrete valuation ring with maximal ideal generated by $p$. From the above we see that $pW(k)=(0, a_0^p, a_1^p, \ldots )$. This clearly gives $W(k)/pW(k)\simeq k$.

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

## Examples

### Basic examples

• $W_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p$ the $p$-adic integers.

• $W_{p^\infty}(\mathbb{F}_{p^n})$ is the unique unramified extension of $\mathbb{Z}_p$ 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 $W(A)$ 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)

### 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 $p \gt 0$ VII“ Math. Ann. , 134 (1957) pp. 114–133

• Michiel Hazewinkel, Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials, Transactions of the AMS (1980)

Surveys incluce

### Modern surveys

Review in the context of the Kummer-Artin-Schreier-Witt exact sequence is in

• Ariane Mézard, Matthieu Romagny, Dajano Tossici, section 2 of Sekiguchi-Suwa theory revisited (arXiv:1104.2222)

### Further development of the theory

In the context of Borger's absolute geometry: