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.

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 ring of symmetric functions, $\Lambda$. The reason is that $\Lambda$ is the free Lambda-ring on the commutative ring $\mathbb{Z}$. Since $\Lambda$ is a Hopf algebra, $W$ 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 ring. 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’‘ $\Phi_n=X^{p^n}+p X^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X$ called Witt polynomials are morphisms

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

of groups.

Definition

We first give the

• Explicit definition in components

and then discuss the

• Universal characterization

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, $\psi(1,0,0,\ldots)=(1,1,1,\ldots)$ 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 $n$-truncation $W_n(A)$ of a Witt ring $W(A)$ is a ring since by definition the first $n$ components of a sum or product in $W(A)$ only depend on the first $n$ components of the elements being added or multiplied. We have $W(A) \simeq lim_n W_n(A)$ and the projection map $W(A)\to W_n(A)$ is a ring homomorphism. We also have the following operations on the truncated Witt rings $W_n(A)$.

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 $W(k)= 1+X k [ [ X] ]$ 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, $W(k)/p^n W(k)\simeq W_n(k)$. 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)

Related concepts

• Witt cohomology

• Lambda-ring

• Hochschild-Witt complex and Kaledin’s non-commutative Witt vectors

• Lubin-Tate ring

• Borger's absolute geometry

• de Rham-Witt complex

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

See also

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

Surveys include:

• eom, Witt vector

• Michel Demazure, Lectures on p-divisible groups (web)

Modern surveys

• Michiel Hazewinkel, Witt vectors, (arXiv)

• Michiel Hazewinkel, Formal Groups and Applications, review in Project Euclid

• H. Lenstra, Construction of the ring of Witt vectors, Eur. J. Math (2019) 5, 1234-–1241. (pdf)

• Richard Pink, Finite group schemes, (pdf)

• Joseph Rabinoff, The theory of Witt vectors, (arXiv)

• Niranjan Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring, (arXiv)

A 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)

Further development of the theory

• A. Dress, C. Siebeneicher, The Burnside ring of profinite groups and the Witt vector construction, Adv. Math., 70, (1988), 87–132.link

• Dmitri Kaledin, non-commutative Witt vectors

• Dmitri Kaledin, universal Witt vectors and the ‘’Japanese cocycle’’, (pdf)

• Lars Hesselholt, Ib Madsen, On the de Rham-Witt complex in mixed characteristic, (pdf)

• Lars Hesselholt, Witt vectors of non-commutative rings and topological cyclic homology, (pdf)

In the context of Borger's absolute geometry:

• James Borger, The basic geometry of Witt vectors, I: The affine case (arXiv)

• James Borger, The basic geometry of Witt vectors, II: Spaces (arXiv)