View source for Witt vector

An element of an algebraic construct, first proposed by E. Witt [[#References|[1]]] in 1936 in the context of the description of unramified extensions of $p$-adic number fields. Witt vectors were subsequently utilized in the study of algebraic varieties over a field of positive characteristic [[#References|[3]]], in the theory of commutative algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $A$ be an associative, commutative ring with unit element. Witt vectors with components in $A$ are infinite sequences $a = (a_0,a_1,\ldots)$, $a_i \in A$, which are added and multiplied in accordance with the following rules:
$$
(a_0,a_1,\ldots) \oplus (b_0,b_1,\ldots) = (S_0(a_0;b_0), S_1(a_0,a_1;b_0,b_1), \ldots)
$$
$$
(a_0,a_1,\ldots) \otimes (b_0,b_1,\ldots) = (M_0(a_0;b_0), M_1(a_0,a_1;b_0,b_1), \ldots)
$$

where $S_n,M_n$ are polynomials in the variables $X_0,\ldots,X_n$, $Y_0,\ldots,Y_n$ with integer coefficients, uniquely defined by the conditions
$$
\Phi_n(S_0,\ldots,S_n) = \Phi_n(X_0,\ldots,X_n) + \Phi_n(Y_0,\ldots,Y_n)
$$
$$
\Phi_n(M_0,\ldots,M_n) = \Phi_n(X_0,\ldots,X_n) \cdot \Phi_n(Y_0,\ldots,Y_n)
$$
where
$$
\Phi_n(Z_0,\ldots,Z_n) = Z_0^{p^n} + p Z_1^{p^{n-1}} + \cdots + p^n Z^n
$$
are polynomials, $n \in \mathbf{N}$ and $p$ is a prime number. In particular,
$$
S_0 = X_0 + Y_0 \ ;\ \ \ S_1 = X_1 + Y_1 - \sum_{i=1}^{p-1} \frac{1}{p} \binom{p}{i} X_0^i Y_0^{p-i}
$$
$$
M_0 = X_0 \cdot Y_0 \ ;\ \ \ M_1 = X_0^p Y_1 + X_1 Y_0^p + p X_1 Y_1 \ .
$$

The Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by $W(A)$. For any natural number $n$  there also exists a definition of the ring $W_n(A)$ of truncated Witt vectors of length $n$. The elements of this ring are finite tuples $a = (a_0,\ldots,a_{n-1})$, $a_i \in A$ with the addition and multiplication operations described above. The canonical mappings
$$
R : W_{n+1}(A) \rightarrow W_n(A)
$$
$$
R : (a_0,\ldots,a_n) \mapsto (a_0,\ldots,a_{n-1})
$$
$$
T : W_n(A) \rightarrow W_{n+1}(A)
$$
$$
T : (a_0,\ldots,a_{n-1}) \mapsto (0,a_0,\ldots,a_{n-1})
$$
are homomorphisms. The rule $A \to W(A)$ (or $A \to W_n(A)$) defines a covariant functor from the category of commutative rings with unit element into the category of rings. This functor may be represented by the ring of polynomials $\mathbf{Z}[X_0,X_1,\ldots]$ (or $\mathbf{Z}[X_0,X_1,\ldots,X_n]$) on which the structure of a ring object has been defined. The spectrum $\mathrm{Spec}\mathbf{Z}[X_0,X_1,\ldots]$ (or $\mathrm{Spec}\mathbf{Z}[X_0,X_1,\ldots,X_n]$) is known as a ''Witt scheme'' (or a ''truncated Witt scheme'') and is a [[ring scheme]] [[#References|[3]]].

Each element $a \in A$ defines a Witt vector
$$
a^T = (a,0,0,\ldots) \in W(A)
$$
called the Teichmüller representative of the element $a$. If $A = k$ is a [[perfect field]] of characteristic $p>0$, then $W(k)$ is a complete [[discrete valuation ring]] of zero characteristic with field of residues $k$ and maximal ideal $pW(k)$. Each element $w \in W(k)$ can be uniquely represented as
$$
w = w_0^T + pw_1^T + p^2 w_2^T + \cdots
$$
where $w_i \in k$. Conversely, each such ring $A$ with field of residues $k = A/(p)$ is canonically isomorphic to the ring $W(k)$. The Teichmüller representation makes it possible to construct a canonical multiplicative homomorphism $k \to W(k)$, splitting the mapping $W(k) \to W(k)/(p)$.

If $k = \mathbf{F}_p$ is the [[prime field]] of $p$ elements, $W(k)$ is the ring of integral $p$-adic numbers $\mathbf{Z}_p$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> E. Witt, "Zyklische Körper und Algebren der characteristik $p$ vom Grad $p^n$. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik $p$" ''J. Reine Angew. Math.'' , '''176''' (1936) pp. 126–140 {{MR|}} {{ZBL|0016.05101}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top"> J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique $p$ VII" ''Math. Ann.'' , '''134''' (1957) pp. 114–133 {{DOI|10.1007/BF01342790}} {{MR|}} {{ZBL|0086.02605}} </TD></TR>
</table>

====Comments====
There is a generalization of the construction above which works for all primes $p$ simultaneously, [[#References|[a3]]]: a [[functor]] $W : \mathsf{Ring} \to \mathsf{Ring}$  called the ''big Witt vector''. Here, $\mathsf{Ring}$ is the [[category]] of commutative, associative rings with unit element. The functor described above, of Witt vectors of infinite length associated to the prime $p$, is a quotient of $W$ which can be conveniently denoted by $W_{p^\infty}$.

For each $n \in \{1,2,\ldots\}$, let $w_n(X)$ be the polynomial
$$
w_n(X) = \sum_{d | n} d X^{n/d} \ .
$$

Then there is the following characterization theorem for the Witt vectors. There is a unique functor $W : \mathsf{Ring} \to \mathsf{Ring}$satisfying the following properties: 1) as a functor $W : \mathsf{Ring} \to \mathsf{Set}$, $W(A) = \{(a_1,a_2,\ldots) : a_i \in A\}$ and $W(\phi)((a_1,a_2,\ldots)) = (\phi(a_1),\phi(a_2),\ldots)$ for any ring homomorphism $\phi : A \to B$; 2) $w_{n,A} : W(A) \to A$, $w_{n,A} : (a_1,a_2,\ldots) \mapsto w_n(a_1,a_2,\ldots)$ is a functorial homomorphism of rings for every $n$ and $A$.

The functor $W$ admits functorial ring endomorphisms $\mathbf{f}_n : W \to W$, for every $n \in \{1,2,\ldots\}$, that are uniquely characterized by $wn \mathbf{f}_m = w_{nm}$ for all $m,n \in \{1,2,\ldots\}$. Finally, there is a functorial homomorphism $\Delta : W({-}) \to W(W({-}))$ that is uniquely characterized by the property $w_{n,W(A)} \Delta_A = \mathbf{f}_{n,A}$ for all $n$,$A$.

To construct $W(A)$, define polynomials $\Sigma_n$; $\Pi_n$; $r_n$ for $n \in \{1,2,\ldots\}$ by the requirements
$$
w_n(\Sigma_1,\ldots,\Sigma_n) = w_n(X) + w_n(Y) \ ;
$$
$$
w_n(\Pi_1,\ldots,\Pi_n) = w_n(X) \cdot w_n(Y) \ ;
$$
$$
w_n(r1,\ldots,r_n) = - w_n(X) \ .
$$

The $\Sigma_n$ and $\Pi_n$ are polynomials in $X_1,\ldots,X_n$; $Y_1,\ldots,Y_n$ and the $r_n$ are polynomials in the $X_1,\ldots,X_n$ and they all have integer coefficients. Now $W(A)$ is defined as the set $W(A) = \{ a = (a_1,a_2,\ldots) : a_i \in A \}$ with operations :
$$
(a_1,a_2,\ldots) + (b_1,b_2,\ldots) = (\Sigma_1(a,b), \Sigma_2(a,b), \ldots) \ ;
$$
$$
(a_1,a_2,\ldots) \cdot (b_1,b_2,\ldots) = (\Pi_1(a,b), \Pi_2(a,b), \ldots) \ ;
$$
$$
- (a_1,a_2,\ldots) = (r_1(a), r_2(a), \ldots) \ .
$$

The zero of $W(A)$ is $(0,0,0,\ldots)$ and the unit element is $(1,0,0,\ldots)$. The Frobenius endomorphisms $\mathbf{f}_n$ and the [[Artin–Hasse exponential]] $\Delta$ are constructed by means of similar considerations, i.e. they are also given by certain universal polynomials. In addition there are the Verschiebung morphisms $\mathbf{V}_n : W({-}) \to W({-})$, which are characterized by
$$
w_n \mathbf{V}_m = \begin{cases} 0 & \text{if}\, n \not\mid m \\ n w_{m/n} & \text{if}\, n \mid m \end{cases} \ .
$$

The $\mathbf{V}_n$ are group endomorphisms of $W(A)$ but not ring endomorphisms.

The ideals $I_n = \{(0,\ldots,0,a_{n+1},a_{n+2},\ldots)\}$ define a topology on $W(A)$ making it a separated complete topological ring.

For each $A \in \mathsf{Ring}$, let $\Lambda(A)$ be the Abelian group $1 + t A[[t]]$ under multiplication of power series;
$$
\bar E : W(A) \rightarrow \Lambda(A)
$$
$$
\bar E : (a_1,a_2,\ldots) \mapsto \prod_{i=1}^\infty \left({ 1 - a_i t^i }\right)
$$
defines a functional isomorphism of Abelian groups, and using the isomorphism $\bar E$ there is a commutative ring structure on $\Lambda(A)$. Using $\bar E$ the Artin–Hasse exponential $\Delta$ defines a functorial homomorphism of rings $W(A) \to \Lambda(W(A))$ making $W(A)$ a functorial special [[Lambda-ring|$\lambda$-ring]]. The Artin–Hasse exponential $\Delta : W \mapsto W \circ W$ defines a cotriple structure on $W$ and the co-algebras for this co-triple are precisely the special $\lambda$-rings (cf. also [[Category]] and [[Triple]]).

On $\Lambda(A)$ the Frobenius and Verschiebung endomorphisms satisfy
$$
\mathbf{f}_n (1 - at) = 1 - a^n t 
$$
$$
V_n(f(t)) = f(t^n) 
$$
and are completely determined by this (plus functoriality and additivity in the case of $\mathbf{f}_n$).

For each supernatural number $\mathbf{n} = (\alpha_p) : \alpha_p \in \{0,1,2,\ldots\} \cup \{\infty\}$,  $p$ prime , one defines $N(\mathbf{n}) = \{ n \in \{1,2,\ldots\} : \nu_p(n) \le \alpha_p \}$, where $\nu_p(n)$ is the $p$-adic valuation of $n$, i.e. the number of prime factors $p$ in $n$. Let
$$
\mathfrak{a}_{\mathbf{n}}(A) = \{ (a_1,a_2,\ldots) : a_i \in A \,,\, a_d = 0 \,\text{for all}\, d \in N(\mathbf{n}) \} \ .
$$

Then $\mathfrak{a}_{\mathbf{n}}(A) $ is an ideal in $W(A)$ and for each supernatural $\mathbf{n}$ a corresponding ring of Witt vectors is defined by
$$
W_{\mathbf{n}}(A) = W(A) / \mathfrak{a}_{\mathbf{n}}(A) \ .
$$

In particular, one thus finds $W_{p^\infty}$, the ring of infinite-length Witt vectors for the prime $p$, discussed in the main article above, as a quotient of the ring of big Witt vectors $W(A)$.

The Artin–Hasse exponential $\Delta : W \to W \circ W$ is compatible in a certain sense with the formation of these quotients, and using also the isomorphism $\bar E$ one thus finds a mapping
$$
\mathbf{Z}_p = W_{p^\infty}(\mathbf{F}_p) \to \Lambda(W_{p^\infty}(\mathbf{F}_p)) = \Lambda(\mathbf{Z}_p)
$$
where $\mathbf{Z}_p$ denotes the $p$-adic integers and $\mathbf{F}_p$ the field of $p$ elements, which can be identified with the classical morphism defined by Artin and Hasse [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].

As an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100164.png" /> is isomorphic to the group of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100165.png" /> of curves in the one-dimensional multiplicative [[Formal group|formal group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100166.png" />. 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, [[#References|[a3]]], [[#References|[a4]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100167.png" /> be the sequence of polynomials with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100168.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100169.png" /></td> </tr></table>

The Cartier ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100170.png" /> is the ring of all formal expressions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100171.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

with the calculation rules

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100173.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100174.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100175.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100176.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png" /></td> </tr></table>

Commutative formal groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100178.png" /> are classified by certain modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100179.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100180.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100181.png" />-algebra, a simpler ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100182.png" /> can be used for this purpose. It consists of all expressions (*) where now the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100183.png" /> only run over the powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100184.png" /> of the prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100185.png" />. The calculation rules are the analogous ones. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100186.png" /> is a perfect field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100187.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100188.png" /> denotes the Frobenius endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100189.png" /> (which in this case is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100191.png" /> can be described as the ring of all expressions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100192.png" /></td> </tr></table>

in two symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100193.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100194.png" /> and with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100195.png" />, with the extra condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100196.png" /> and the calculation rules

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100197.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100198.png" /></td> </tr></table>

This ring, and also its subring of all expressions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100199.png" /></td> </tr></table>

is known as the Dieudonné ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100200.png" /> and certain modules (called Dieudonné modules) over it classify unipotent commutative affine group schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100201.png" />, cf. [[#References|[a5]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der $\ell$-ten Potenzreste im Körper der $\ell$-ten Einheitswurzeln" ''Abh. Math. Sem. Univ. Hamburg'' , '''6''' (1928) pp. 146–162 {{MR|}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Whaples, "Generalized local class field theory III: Second form of the existence theorem, structure of analytic groups" ''Duke Math. J.'' , '''21''' (1954) pp. 575–581 {{MR|73645}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Hazewinkel, "Twisted Lubin–Tate formal group laws, ramified Witt vectors and (ramified) Artin–Hasse exponentials" ''Trans. Amer. Math. Soc.'' , '''259''' (1980) pp. 47–63 {{MR|0561822}} {{ZBL|0437.13014}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) {{MR|506881}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR>
</table>

{{TEX|part}}