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ring of Witt vectors

Disambiguation

There is also a different notion of Witt group? and Witt ring. This article is about of groups of Witt vectors and rings of Witt vectors; however these are often called just Witt groups and Witt rings, too.

Idea

A Witt vector is an infinite sequence of elements of a commutative ring $k$. There is a unique 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 pime 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.

A Witt ring is in particular a Lambda-ring and the assignment $W:k\mapsto W(k)$ of a commutative ring to its Witt ring is a functor which has a left adjoint ”forgetting the $\lambda$-structure”. More over $W$ 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 $W(A)$ 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 to non-commutative Witt vectors, however these only carry a group- but no ring structure.

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 $T_p:=\{x\mid x^{p-1}=1\}\cup \{0\}$. 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” ${\Phi }_n=X^{p^n}+{\mathrm{pX}}^{p^{n-1}}+p^2{X}^{p^{n-2}}+\dots +p^{n}X$ called Witt polynomials are morphisms

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

of groups.

Definition

Definition and Theorem

Let $k$ be a commutaive ring.

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

$$(a_1,a_2,\dots ,)+(b_1,b_2,\dots ):=({\Sigma }_1(a_1,b_1),{\Sigma }_2(a_1,a_2,b_1,b_2),\dots )$$(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

$$(a_1,a_2,\dots )\cdot (b_1,b_2,\dots ):=({\Pi }_1(a_1,b_1),{\Pi }_2(a_1,a_2,b_1,b_2),\dots )$$(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 ${\Sigma }_i$ (described below) are called addition polynomials and the ${\Pi }_i$ (described below) are called multiplication polynomials

The assignment $W:k\mapsto W(k)$ is a functor. This functor is the unique endofunctor $W:\mathrm{CRing}\to \mathrm{CRing}$ such that all Witt polynomials

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

are morphisms of rings. In fact this unicity indicates that $W$ has a left adjoint.

The Witt polynomials, the ${\Sigma }_i$, the ${\Pi }_i$, phantom components

let $x_1,x_2,\dots$ be a collection of indeterminate?s. We can define an infinite collection of polynomials in $\mathbb{Z}[x_1,x_2,\dots ]$ 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 ${\displaystyle w_n(X)=\sum _{d\mid n}{\mathrm{dx}}_n^{n/d}}$. 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 $\varphi (z_1,z_2)\in \mathbb{Z}[z_1,z_2]$. This just an arbitrary two variable polynomial with coefficients in $\mathbb{Z}$.

We can define new polynomials ${\Phi }_i(x_1,\dots x_i,y_1,\dots y_i)$ such that the following condition is met $\varphi (w_n(x_1,\dots ,x_n),w_n(y_1,\dots ,y_n))=w_n({\Phi }_1(x_1,y_1),\dots ,{\Phi }_n(x_1,\dots x_n,y_1,\dots ,y_n))$.

In short we’ll notate this $\varphi (w_n(X),w_n(Y))=w_n(\Phi (X,Y))$. 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 $(a_1,a_2,\dots )$ with entries in $A$. Then we define addition and multiplication by $(a_1,a_2,\dots ,)+(b_1,b_2,\dots )=({\Sigma }_1(a_1,b_1),{\Sigma }_2(a_1,a_2,b_1,b_2),\dots )$ and $(a_1,a_2,\dots )\cdot (b_1,b_2,\dots )=({\Pi }_1(a_1,b_1),{\Pi }_2(a_1,a_2,b_1,b_2),\dots )$.

Proof of the theorem

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 $(a_1,a_2,\dots )\mapsto (w_1(a),w_2(a),\dots )$. 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, $w(0,0,\dots )=(0,0,\dots )$, so $(0,0,\dots )$ is the additive identity, $W(1,0,0,\dots )=(1,1,1,\dots )$ which shows $(1,0,0,\dots )$ is the multiplicative identity, and $w({\iota }_1(a),{\iota }_2(a),\dots )=(-a_1,-a_2,\dots )$, so we see $({\iota }_1(a),{\iota }_2(a),\dots )$ 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 $(0,0,\dots )\mapsto (0,0,\dots )$ and $(1,0,0,\dots )\mapsto (1,0,0,\dots )$ 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.

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 $\mathrm{VR}=\mathrm{RV}$ where $R$ is the restriction map.

The restriction map

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

The Frobenius morphism

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

Duality of finite Witt groups

For a $k$-ring $R$ let $W^{\prime }(R)$ 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

$$E(-,1):\{\begin{array}{l}{W}^{\prime }\to {\mu }_k\\ w\mapsto E(w,1)\end{array}$$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) $W^{\prime }(R)$ is an ideal in $W(R)$.

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

c) The morphism

$$\{\begin{array}{l}W\times W^{\prime }\to {\mu }_k\\ (x,y)\mapsto E(\mathrm{xy},1)\end{array}$$\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

$$W^{\prime }\to D(W)$$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 $E(\mathrm{xy},1)\in R^{\times }$ and

$$E(V^{n}xy,1)=E(T^n(x{F}^{n}y),1)=E(x{F}^{n}y,1)$$E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1)

Definition and Theorem

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

$${\sigma }_n:\{\begin{array}{l}{W}_{\mathrm{nk}}\to W_k\\ ({\alpha }_0,\dots ,{\alpha }_{n-1})\mapsto ({\alpha }_0,\dots ,{\alpha }_{n-1},0,\dots )\end{array}$$\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 $\ker (F_n^m)$ in $W^{\prime }$. Note that ${\sigma }_n$ is not a morphism of groups.

We define the bilinear map

$$<-,->:\{\begin{array}{l}\ker (F_n^m)\times \ker (F_m^n)\to R^{\times }\\ <x,y>\mapsto E({\sigma }_n(x){\sigma }_m(y),1)\end{array}$$\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 $<x,y>$ is bilinear and gives an isomorphism

$$\ker (F_n^m)\simeq D(\ker (F_m^n)$$ker(F^m_n)\simeq D(ker(F^n_m)

and satisfies

$$<x,ty>=<fx,y>$$\lt x,t y\gt=\lt f x,y\gt

$$<x,ry>=<ix,y>$$\lt x,r y\gt=\lt i x,y\gt

where the morphisms are

$$\begin{array}{ccc}\ker (F_n^m)& \stackrel{t}{\to }& \ker (F_{n+1}^m)\\ {\downarrow }^f& & {\downarrow }^r\\ \ker (F_n^{m-1})& \stackrel{i}{\hookrightarrow }& \ker (F_n^m)\end{array}$$\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 $\ker (F_m^n)$ 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 $W(k)=1+k[[X]]$ 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 $\mathrm{pW}(k)=(0,a_0^p,a_1^p,\dots )$. This clearly gives $W(k)/\mathrm{pW}(k)\simeq k$.

Also, $W(k)/p^{\mathrm{nW}}(k)\simeq W_n(k)$. 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

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

• $W_{p^{\mathrm{\infty }}}({𝔽}_{p^n})$ is the unique unramified extension? of ${\mathbb{Z}}_p$ of degree $n$.

Related Concepts

• Witt Cohomology

• Lambda-ring

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

References

Original texts and classical surveys

• Michel Demazure, lectures on p-divisible groups web

• Ernst Witt, Zyklische Körper und Algebren der Characteristik p vom Grad $p^n$. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik $p^n$, web

Modern surveys

• Michiel Hazewinkel, Formal Groups and Applications, review in projecteuclid

• Hazewinkel, Witt vectors, pdf.

• Joseph Rabinoff, the theory of Witt vectors, pdf

• Richard Pink, finite group schemes, pdf

Further development of the theory

• 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 comples in mixed characteristic, pdf

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