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.
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 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.
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
of groups.
We first give the
and then discuss the
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
and the multiplication
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.
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 $\displaystyle w_n(X)=\sum_{d|n} dx_n^{n/d}$. 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 $\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]$. 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.
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 $(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$.
The assignment
is a functor
from the category of commutative rings to that of Lambda-rings.
Composed with the forgetful functor
this is the unique endofunctor $W \;\colon\; CRing \longrightarrow CRing$ such that all Witt polynomials
are homomorphisms of rings.
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.
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
Hence rings of Witt-vectors are the co-free Lambda-rings.
This statement appears in (Hazewinkel 08, p. 87, p. 97).
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).
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.
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 $R: W_{n+1}(A)\to W_n(A)$ is given by $(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})$.
The Frobenius 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.
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
is a morphism of group schemes to the multiplicative group scheme $\mu_k$.
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
is bilinear and gives an isomorphism of group schemes
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
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
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
then $\lt x,y\gt$ is bilinear and gives an isomorphism
and satisfies
where the morphisms are
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
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 $1$.
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.
$W_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p$ the $p$-adic integers.
$W_{p^\infty}(\mathbb{F}_{p^n})$ is the unique unramified extension of $\mathbb{Z}_p$ of degree $n$.
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)
Hochschild-Witt complex and Kaledin’s non-commutative Witt vectors
witt vectors were introduced in
In the context of formal group laws they were used in
See also
Surveys incluce
Michiel Hazewinkel, Formal Groups and Applications, review in projecteuclid
Joseph Rabinoff, The theory of Witt vectors (pdf)
Richard Pink, finite group schemes, pdf
Review in the context of the Kummer-Artin-Schreier-Witt exact sequence is in
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
In the context of Borger's absolute geometry:
James Borger, The basic geometry of Witt vectors, I: The affine case (arXiv:0801.1691)
James Borger, The basic geometry of Witt vectors, II: Spaces (arXiv:1006.0092)