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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:
+
{{TEX|done}}
$$
 
(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
+
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 $-
$$
+
adic number fields. Witt vectors were subsequently utilized in the study of algebraic varieties over a field of positive characteristic [[#References|[3]]], in the theory of commutative algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let  $ A $
\Phi_n(S_0,\ldots,S_n) = \Phi_n(X_0,\ldots,X_n) + \Phi_n(Y_0,\ldots,Y_n)
+
be an associative, commutative ring with unit element. Witt vectors with components in  $ A $
$$
+
are infinite sequences  $ a = (a _{0} , a _{1} , .  .  . ) $,  
$$
+
$ a _{i} \in A $,
\Phi_n(M_0,\ldots,M_n) = \Phi_n(X_0,\ldots,X_n) \cdot \Phi_n(Y_0,\ldots,Y_n)
+
which are added and multiplied in accordance with the following rules: $$  
$$
+
(a _{0} ,\ a _{1} ,\dots ) \dot{+}
where
+
(b _{0} ,\ b _{1} ,\dots )  =
$$
 
\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,
 
 
$$
 
$$
S_0 = X_0 + Y_0 \ ;\ \ \ S_1 = X_1 + Y_1 - \sum_{i=1}^{p-1} \frac{1}{p} \binom{p}{i} X_0^i Y_0^{p-i}
+
$$
 +
=
 +
(S _{0} (a _{0} ,\  b _{0} ),\ S _{1} (a _{0} ,\ a _{1} ; \  b _{0} ,\  b _{1} ) ,  .  .  . ),
 
$$
 
$$
 +
$$
 +
(a _{0} ,\  a _{1} ,  .  .  . ) \dot \times  (b _{0} ,\  b _{1} ,  .  .  . )  =
 
$$
 
$$
M_0 = X_0 \cdot Y_0 \ ;\ \ \ M_1 = X_0^p Y_1 + X_1 Y_0^p + p X_1 Y_1 \ .
+
$$
 +
=
 +
(M _{0} (a _{0} ,\ b _{0} ),\ M _{1} (a _{0} ,\  a _{1} ; \ b _{0} ,\ b _{1} ) ,  .  .  . ),
 
$$
 
$$
 
+
where  $ S _{n} $,
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
+
$ M _{n} $
 +
are polynomials in the variables $ X _{0} \dots X _{n} $,
 +
$ Y _{0} \dots Y _{n} $
 +
with integer coefficients, uniquely defined by the conditions $$  
 +
\Phi _{n} (S _{0} \dots S _{n} )  =
 +
\Phi _{n} (X _{0} \dots X _{n} ) +
 +
\Phi _{n} (Y _{0} \dots Y _{n} ),
 
$$
 
$$
R : W_{n+1}(A) \rightarrow W_n(A)
+
$$
 +
\Phi _{n} (M _{0} \dots M _{n} )   =  \Phi _{n} (X _{0} \dots X _{n} ) \cdot \Phi _{n} (Y _{0} \dots Y _{n} );
 
$$
 
$$
 +
where $$
 +
\Phi _{n}  =  Z _{0} ^ {p ^ n} + pZ _{1} ^ {p ^ n-1} + \dots
 +
+ 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} ;
 
$$
 
$$
R : (a_0,\ldots,a_n) \mapsto (a_0,\ldots,a_{n-1})
+
$$
 +
M _{0}  =  X _{0} Y _{0} ,   M _{1}  =  X
 +
_{0} ^{p} Y _{1} + X _{1} Y _{0} ^{p} + pX _{1} Y _{1} .
 
$$
 
$$
 +
The Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by  $  W(A) $.
 +
For any natural number  $  n $
 +
there also exists a definition of the ring  $  W _{n} (A) $
 +
of truncated Witt vectors of length  $  n $.
 +
The elements of this ring are finite tuples  $  a = (a _{0} \dots a _{n-1} ) $,
 +
$  a _{i} \in A $,
 +
with the addition and multiplication operations described above. The canonical mappings $$
 +
R: \  W _{n+1} (A)  \rightarrow  W _{n} (A),
 
$$
 
$$
T : W_n(A) \rightarrow W_{n+1}(A)
+
$$
 +
R ((a _{0} \dots a _{n} ))  =  (a _{0} \dots a _{n-1} ) ,
 
$$
 
$$
 +
$$
 +
T: \  W _{n} (A)  \rightarrow  W _{n+1} (A),
 
$$
 
$$
T : (a_0,\ldots,a_{n-1}) \mapsto (0,a_0,\ldots,a_{n-1})
+
$$
 +
T ((a _{0} \dots a _{n-1} ))  =  (0,\ a _{0} \dots 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]]].
+
are homomorphisms. The rule $ A \mapsto W(A) $(
 +
or $ A \mapsto 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} \dots X _{n} ,\dots ] $(
 +
or $ \mathbf Z [X _{0} \dots X _{n-1} ] $)  
 +
on which the structure of a ring object has been defined. The spectrum $   \mathop{\rm Spec}\nolimits \  \mathbf Z [X _{0} \dots X _{n} ,\dots ] $(
 +
or $   \mathop{\rm Spec}\nolimits \  \mathbf Z [X _{0} \dots X _{n-1} ] $)  
 +
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
+
Each element $ a \in A $
 +
defines a Witt vector $$
 +
a ^ \tau    =  (a,\  0,\  0 ,  .  .  . )  \in  W \  (A),
 
$$
 
$$
a^T = (a,0,0,\ldots) \in W(A)
+
called the Teichmüller representative of the element  $  a $.
 +
If  $  A = k $
 +
is a perfect field of characteristic  $  p > 0 $,  
 +
$  W(k) $
 +
is a complete discrete valuation ring of zero characteristic with field of residues  $  k $
 +
and maximal ideal  $  pW(k) $.
 +
Each element  $  \omega \in W(k) $
 +
can be uniquely represented as $$
 +
\omega  =  \omega _{0} ^ \tau  +
 +
p \omega _{1} ^ \tau  + p ^{2} \omega _{2} ^ \tau  + \dots ,
 
$$
 
$$
called the Teichmüller representative of the element $a$. If $A = k$ is a [[perfect field]] of characteristic $p>0$, then $W(k)$ is a complete [[discrete valuation ring]] of zero characteristic with field of residues $k$ and maximal ideal $pW(k)$. Each element $w \in W(k)$ can be uniquely represented as
+
where  $ \omega _{i} \in k $.  
 +
Conversely, each such ring  $ A $
 +
with field of residues  $ k = A/p $
 +
is canonically isomorphic to the ring  $ W(k) $.
 +
The Teichmüller representation makes it possible to construct a canonical multiplicative homomorphism  $ k \rightarrow W(k) $,
 +
splitting the mapping $$
 +
W (k)   \rightarrow  W (k) / p  \simeq  k.
 
$$
 
$$
w = w_0^T + pw_1^T + p^2 w_2^T + \cdots
+
If  $  k = \mathbf F _{p} $
$$
+
is the prime field of  $ p $
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 $W(k) \to W(k)/(p)$.
+
elements, $ W( \mathbf F _{p} ) $
 +
is the ring of integral  $ p $-
 +
adic numbers  $ \mathbf Z _{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====
 
====References====
<table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Witt, "Zyklische Körper und Algebren der characteristik <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810058.png" /> vom Grad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810059.png" />. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810060.png" />" ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810061.png" /> VII" ''Math. Ann.'' , '''134''' (1957) pp. 114–133 {{MR|}} {{ZBL|}} </TD></TR></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====
 
====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 $W_{p^\infty}$.
+
There is a generalization of the construction above which works for all primes $ p $
 +
simultaneously, [[#References|[a3]]]: a functor $ W : \ \mathbf{Ring} \rightarrow \mathbf{Ring} $
 +
called the big Witt vector. Here, $ \mathbf{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
+
 
$$
+
For each $ n \in \{ 1,2,\dots \} $,  
w_n(X) = \sum_{d | n} d X^{n/d} \ .
+
let $ w _{n} (X) $
 +
be the polynomial $$  
 +
w _{n} (X)   =   \sum _{d\mid n} dX _{d} ^{n/d} .
 
$$
 
$$
 +
Then there is the following characterization theorem for the Witt vectors. There is a unique functor  $  W : \  \mathbf{Ring} \rightarrow \mathbf{Ring} $
 +
satisfying the following properties: 1) as a functor  $  W: \  \mathbf{Ring} \rightarrow  \mathop{\rm Set}\nolimits $,
 +
$  W (A) = \{ {(a _{1} ,\  a _{2} , \dots )} : {a _{i} \in A} \} $
 +
and  $  W ( \phi ) (a _{1} ,\  a _{2} ,  .  .  ) = ( \phi (a _{1} ) ,\  \phi (a _{2} ) ,\dots ) $
 +
for any ring homomorphism  $  \phi : \  A \rightarrow B $;
 +
2)  $  w _ {n , A} : \  W(A) \rightarrow A $,
 +
$  ( a _{1} ,\  a _{2} ,\dots ) \mapsto w _{n} (a _{1} ,\  a _{2} ,\dots ) $
 +
is a functorial homomorphism of rings for every  $  A $
 +
and  $  n \in \{ 1,\  2,\dots \} $.
 +
  
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 \rightarrow W $,
 +
for every  $  n \in \{ 1,\  2,\dots \} $,
 +
that are uniquely characterized by  $ w _{n} \mathbf f _{m} = w _{nm} $
 +
for all  $  n,\  m \in \{ 1,\  2,\dots \} $.
 +
Finally, there is a functorial homomorphism  $ \Delta : \  W(-) \rightarrow W(W(-)) $
 +
that is uniquely characterized by the property  $ w _ {n, W(A)} \Delta _{A} = \mathbf f _ {n, A} $
 +
for all  $ n $,
 +
$ 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
+
To construct $ W(A) $,  
 +
define polynomials $ \Sigma _{1} \dots \Sigma _{n} ,\dots $;  
 +
$ \Pi _{1} \dots \Pi _{n} ,\dots $;  
 +
$  r _{1} \dots r _{n} ,\dots $
 +
by the requirements $$  
 +
w _{n} ( \Sigma _{1} \dots \Sigma _{n} )  = 
 +
w _{n} (X) + w _{n} (Y),
 
$$
 
$$
w_n(\Sigma_1,\ldots,\Sigma_n) = w_n(X) + w_n(Y) \ ;
+
$$
 +
w _{n} ( \Pi _{1} \dots \Pi _{n} )   =   w _{n} (X) w _{n} (Y),
 
$$
 
$$
 +
$$
 +
w _{n} ( r _{1} \dots r _{n} )  =  - w _{n} ( X) .
 
$$
 
$$
w_n(\Pi_1,\ldots,\Pi_n) = w_n(X) \cdot w_n(Y) \ ;
+
The  $  \Sigma _{n} $
 +
and  $  \Pi _{n} $
 +
are polynomials in  $  X _{1} \dots X _{n} $;
 +
$  Y _{1} \dots Y _{n} $
 +
and the  $  r _{n} $
 +
are polynomials in the  $  X _{1} \dots X _{n} $
 +
and they all have integer coefficients.  $  W(A) $
 +
is now defined as the set  $  W(A) = \{ {\mathbf a = (a _{1} ,\ a _{2} ,\dots )} : {a _{i} \in A} \} $
 +
with addition, multiplication and "minus" : $$
 +
(a _{1} ,\  a _{2} ,\dots ) +
 +
(b _{1} ,\  b _{2} ,\dots )  =
 +
( \Sigma _{1} ( \mathbf a ) ,\ \Sigma _{2} ( \mathbf a ) ,\dots )
 
$$
 
$$
 +
$$
 +
(a _{1} ,\  a _{2} ,\dots ) (b _{1} ,\  b _{2} ,\dots )
 +
  =  ( \Pi _{1} ( \mathbf a ) ,\  \Pi _{2} ( \mathbf a ) ,\dots ) -
 
$$
 
$$
w_n(r1,\ldots,r_n) = - w_n(X) \ .
+
$$
 +
-
 +
(a _{1} ,\ a _{2} ,\dots )   =   ( r _{1} ( \mathbf a ) ,\  r _{2} ( \mathbf a ) ,\dots ) .
 
$$
 
$$
 +
The zero of  $  W(A) $
 +
is  $  ( 0,\  0 ,\dots ) $
 +
and the unit element is  $  ( 1,\  0 ,\  0 ,\dots ) $.
 +
The Frobenius endomorphisms  $  \mathbf f _{n} $
 +
and the Artin–Hasse exponential  $  \Delta $
 +
are constructed by means of similar considerations, i.e. they are also given by certain universal polynomials. In addition there are the Verschiebung morphisms  $  \mathbf V _{n} : \  W(-) \rightarrow W(-) $,
 +
which are characterized by $$
 +
w _{m} \mathbf V _{n}  =  \left \{
  
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 :
+
\begin{array}{ll}
 +
0  &  \textrm{ if }  n  \textrm{ does  not  divide  } m,  \\
 +
nw _{m/n}  &  \textrm{ if }  n  \textrm{ divides }  m.  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
The  $ \mathbf V _{m} $
 +
are group endomorphisms of  $ W(-) $
 +
but not ring endomorphisms.
 +
 
 +
The ideals  $ I _{n} = \{ ( 0 \dots 0,\ a _{n+1} ,\  a _{n+2} ,\dots ) \} \subset W(A) $
 +
define a topology on  $ W(A) $
 +
making  $ W(A) $
 +
a separated complete topological ring.
 +
 
 +
For each  $  A \in \mathbf{Ring} $,  
 +
let  $  \Lambda (A) $
 +
be the Abelian group  $  1 + t A [[t]] $
 +
under multiplication of power series; $$
 +
\overline{E}\; : \ W(A\rightarrow  \Lambda (A),
 
$$
 
$$
(a_1,a_2,\ldots) + (b_1,b_2,\ldots) = (\Sigma_1(a,b), \Sigma_2(a,b), \ldots) \ ;
+
$$
 +
( a _{1} ,\  a _{2} ,\dots )   \mapsto  \prod _{i=1} ^ \infty (1- a _{i} t ^{i} ) ,
 
$$
 
$$
 +
defines a functional isomorphism of Abelian groups, and using the isomorphism  $  \overline{E}\; $
 +
there is a commutative ring structure on  $  \Lambda (A) $.
 +
Using  $  \overline{E}\; $
 +
the Artin–Hasse exponential  $  \Delta $
 +
defines a functorial homomorphism of rings $$
 +
W(A)  \rightarrow  \Lambda (W(A))
 
$$
 
$$
(a_1,a_2,\ldots) \cdot (b_1,b_2,\ldots) = (\Pi_1(a,b), \Pi_2(a,b), \ldots) \ ;
+
making  $  W(A) $
$$
+
a functorial special [[Lambda-ring| $  \lambda $-
$$
+
ring]]. The Artin–Hasse exponential  $  \Delta : \ W \rightarrow W \circ W $
- (a_1,a_2,\ldots) = (r_1(a), r_2(a), \ldots) \ .
+
defines a cotriple structure on  $ W $
$$
+
and the co-algebras for this co-triple are precisely the special  $ \lambda $-
 +
rings (cf. also [[Category|Category]] and [[Triple|Triple]]).
  
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
+
On  $ \Lambda (A) $
 +
the Frobenius and Verschiebung endomorphisms satisfy $$  
 +
\mathbf f _{n} (1-at)   =  (1-a ^{n} t) ,
 
$$
 
$$
w_n \mathbf{V}_m = \begin{cases} 0 & \text{if}\, n \not\mid m \\ n w_{m/n} & \text{if}\, n \mid m \end{cases} \ .
+
$$
 +
\mathbf V _{n} f(t)  =   f(t ^{n} ) ,
 
$$
 
$$
 +
and are completely determined by this (plus functoriality and additivity in the case of  $  \mathbf f _{n} $).
  
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;
+
For each supernatural number  $ \mathbf n = \prod _{p} p ^ {\alpha _ p} $,
 +
$  \alpha _{p} \in \{ 0,\  1,\  2,\dots \} \cup \{ \infty \} $,  
 +
one defines  $ N ( \mathbf n ) = \{ {n \in \{ 1,\  2,\dots \}} : {v _{p} (n) \leq \alpha _{p  } \textrm{ for  all  "prime"  numbers }  p} \} $,
 +
where  $  v _{p} $
 +
is the $ p $-
 +
adic valuation of $  n $,
 +
i.e. the number of prime factors  $  p $
 +
in  $  n $.
 +
Let $$
 +
\mathfrak a _ {\mathbf n} (A)  =
 
$$
 
$$
\bar E : W(A) \rightarrow \Lambda(A)
+
$$
 +
 +
\{ {(a _{1} ,\  a _{2} ,\dots )} : {
 +
a _{d} = 0  \textrm{ 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) .
 
$$
 
$$
\bar E : (a_1,a_2,\ldots) \mapsto \prod_{i=1}^\infty \left({ 1 - a_i t^i }\right)
+
In particular, one thus finds  $  W _ {p ^ \infty} (A) $,
 +
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 \rightarrow W \circ W $
 +
is compatible in a certain sense with the formation of these quotients, and using also the isomorphism  $  \overline{E}\; $
 +
one thus finds a mapping $$
 +
\mathbf Z _{p}  =   W _ {p ^ \infty} ( \mathbf F _{p} )  \rightarrow 
 +
\Lambda (W _ {p ^ \infty} ( \mathbf F _{p} ) )   = 
 +
\Lambda ( \mathbf Z _{p} ) ,
 
$$
 
$$
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]]).
+
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]]].
  
On $\Lambda(A)$ the Frobenius and Verschiebung endomorphisms satisfy
+
As an Abelian group  $ W(A) $
 +
is isomorphic to the group of curves  $  {\mathcal C} ( \mathbf G _{m} ; \  A) $
 +
of curves in the one-dimensional multiplicative [[Formal group|formal group]]  $  \mathbf G _{m} $.
 +
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  $  r _{n} (X,\  Y) $
 +
be the sequence of polynomials with coefficients in  $  \mathbf Z $
 +
defined by $$
 +
X ^{n} + Y ^{n}  =  \sum _{d\mid n} d r _{d} (X,\  Y) ^{n/d} .
 
$$
 
$$
\mathbf{f}_n (1 - at) = 1 - a^n t
+
The Cartier ring  $  \mathop{\rm Cart}\nolimits (A) $
 +
is the ring of all formal expressions $$ \tag{*}
 +
\sum _ {i,j \in \{ 1, 2,\dots \}}
 +
\mathbf V _{i} \langle  a _{ij} \rangle \mathbf f _{j}  $$
 +
with the calculation rules $$
 +
\langle a><b\rangle  =  \langle ab\rangle , 
 +
\langle 1\rangle  =  \mathbf f _{1}   =  \mathbf V _{1=
 +
\textrm{ unit  element  } 1 ,
 
$$
 
$$
 +
$$
 +
\mathbf V _{n} \mathbf V _{m}  =  \mathbf V _{nm} , 
 +
\mathbf f _{n} \mathbf f _{m}  =  \mathbf f _{nm} ,
 
$$
 
$$
V_n(f(t)) = f(t^n)
+
$$
 +
\langle a\rangle \mathbf V _{m}  =  \mathbf V _{m} \langle  a ^{m} \rangle , 
 +
\mathbf f _{m} \langle a\rangle  =   \langle  a ^{m} \rangle \mathbf f _{m} ,
 
$$
 
$$
and are completely determined by this (plus functoriality and additivity in the case of $\mathbf{f}_n$).
+
$$
 
+
\mathbf V _{m} \mathbf f _{n}   =   \mathbf f _{n} \mathbf V _{m}   \textrm{ if }   (n,\ m) = 1 ,
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
+
$$
 +
$$  
 +
\mathbf f _{n} \mathbf V _{n}  =  1 + \dots + 1   ( n   \textrm{ summands } ) ,
 
$$
 
$$
\mathfrak{a}_{\mathbf{n}}(A) = \{ (a_1,a_2,\ldots) : a_i \in A \,,\, a_d = 0 \,\text{for all}\, d \in N(\mathbf{n}) \} \ .
+
$$
 +
\langle a+b\rangle  =  \sum _{n=1} ^ \infty \mathbf V _{n} \langle  r _{n} ( a,\ b) \rangle \mathbf f _{n} .
 
$$
 
$$
 
+
Commutative formal groups over  $  A $
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
+
are classified by certain modules over  $   \mathop{\rm Cart}\nolimits (A) $.
 +
In case  $  A $
 +
is a $  \mathbf Z _{(p)} $-
 +
algebra, a simpler ring  $  \mathop{\rm Cart}\nolimits _{p} (A) $
 +
can be used for this purpose. It consists of all expressions (*) where now the  $  i,\  j $
 +
only run over the powers  $  p ^{0} ,\  p ^{1} ,\  p ^{2} ,  .  .  . $
 +
of the prime  $  p $.
 +
The calculation rules are the analogous ones. In case  $  k $
 +
is a perfect field of characteristic  $  p > 0 $
 +
and  $  \sigma $
 +
denotes the Frobenius endomorphism of  $  W(k) $(
 +
which in this case is given by  $ \sigma ( a _{1} ,\  a _{2} ,  .  .  . ) = ( a _{1} ^{p} ,\  a _{2} ^{p} ,  .  .  ) $),
 +
then  $   \mathop{\rm Cart}\nolimits _{p} (k) $
 +
can be described as the ring of all expressions $$
 +
x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} +
 +
\sum _{j=1} ^ \infty y _{j} \mathbf f ^{i} ,
 
$$
 
$$
W_{\mathbf{n}}(A) = W(A) / \mathfrak{a}_{\mathbf{n}}(A) \ .
+
in two symbols  $  \mathbf f $
 +
and  $  \mathbf V $
 +
and with coefficients in  $  W _ {p ^ \infty} (k) $,
 +
with the extra condition  $  \mathop{\rm lim}\nolimits _ {i \rightarrow \infty} \  y _{i} = 0 $
 +
and the calculation rules $$
 +
\mathbf f x  =  \sigma (x) \mathbf f , 
 +
\mathbf V x  =  \sigma ^{-1} (x) \mathbf V ,
 
$$
 
$$
 
+
$$  
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 $W(A)$.
+
\mathbf f \mathbf V  =  \mathbf V \mathbf f  =  p .
 
 
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
 
 
$$
 
$$
\mathbf{Z}_p = W_{p^\infty}(\mathbf{F}_p) \to \Lambda(W_{p^\infty}(\mathbf{F}_p)) = \Lambda(\mathbf{Z}_p)
+
This ring, and also its subring of all expressions $$
 +
x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} +
 +
\sum _{j=1} ^ {< \infty} y _{j} \mathbf f ^{j} ,
 
$$
 
$$
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]]].
+
is known as the Dieudonné ring $  D(k) $
 
+
and certain modules (called Dieudonné modules) over it classify unipotent commutative affine group schemes over $  k $,  
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]]].
+
cf. [[#References|[a5]]].
 
 
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====
 
====References====
<table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100202.png" />-ten Potenzreste im Körper der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100203.png" />-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>
<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>
 
 
 
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Latest revision as of 18:25, 22 December 2019


An element of an algebraic construct, first proposed by E. Witt [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 [3], in the theory of commutative algebraic groups [4], [5], and in the theory of formal groups [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} , . . . ) $, $ a _{i} \in A $, which are added and multiplied in accordance with the following rules: $$ (a _{0} ,\ a _{1} ,\dots ) \dot{+} (b _{0} ,\ b _{1} ,\dots ) = $$ $$ = (S _{0} (a _{0} ,\ b _{0} ),\ S _{1} (a _{0} ,\ a _{1} ; \ b _{0} ,\ b _{1} ) , . . . ), $$ $$ (a _{0} ,\ a _{1} , . . . ) \dot \times (b _{0} ,\ b _{1} , . . . ) = $$ $$ = (M _{0} (a _{0} ,\ b _{0} ),\ M _{1} (a _{0} ,\ a _{1} ; \ b _{0} ,\ b _{1} ) , . . . ), $$ where $ S _{n} $, $ M _{n} $ are polynomials in the variables $ X _{0} \dots X _{n} $, $ Y _{0} \dots Y _{n} $ with integer coefficients, uniquely defined by the conditions $$ \Phi _{n} (S _{0} \dots S _{n} ) = \Phi _{n} (X _{0} \dots X _{n} ) + \Phi _{n} (Y _{0} \dots Y _{n} ), $$ $$ \Phi _{n} (M _{0} \dots M _{n} ) = \Phi _{n} (X _{0} \dots X _{n} ) \cdot \Phi _{n} (Y _{0} \dots Y _{n} ); $$ where $$ \Phi _{n} = Z _{0} ^ {p ^ n} + pZ _{1} ^ {p ^ n-1} + \dots + p ^{n} Z _{n} $$ are polynomials, $ n \in \mathbf N $ and $ p $ is a prime number. In particular, $$ S _{0} = X _{0} + Y _{0} ; S _{1} = X _{1} + Y _{1} - \sum _ {i = 1} ^ {p-1} { \frac{1}{p} } \binom{p}{i} X _{0} ^{i} Y _{0} ^{p-i} ; $$ $$ M _{0} = X _{0} Y _{0} , M _{1} = X _{0} ^{p} Y _{1} + X _{1} Y _{0} ^{p} + pX _{1} Y _{1} . $$ 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} \dots a _{n-1} ) $, $ a _{i} \in A $, with the addition and multiplication operations described above. The canonical mappings $$ R: \ W _{n+1} (A) \rightarrow W _{n} (A), $$ $$ R ((a _{0} \dots a _{n} )) = (a _{0} \dots a _{n-1} ) , $$ $$ T: \ W _{n} (A) \rightarrow W _{n+1} (A), $$ $$ T ((a _{0} \dots a _{n-1} )) = (0,\ a _{0} \dots a _{n-1} ), $$ are homomorphisms. The rule $ A \mapsto W(A) $( or $ A \mapsto 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} \dots X _{n} ,\dots ] $( or $ \mathbf Z [X _{0} \dots X _{n-1} ] $) on which the structure of a ring object has been defined. The spectrum $ \mathop{\rm Spec}\nolimits \ \mathbf Z [X _{0} \dots X _{n} ,\dots ] $( or $ \mathop{\rm Spec}\nolimits \ \mathbf Z [X _{0} \dots X _{n-1} ] $) is known as a Witt scheme (or a truncated Witt scheme) and is a ring scheme [3].

Each element $ a \in A $ defines a Witt vector $$ a ^ \tau = (a,\ 0,\ 0 , . . . ) \in W \ (A), $$ called the Teichmüller representative of the element $ a $. If $ A = k $ is a perfect field of characteristic $ p > 0 $, $ W(k) $ is a complete discrete valuation ring of zero characteristic with field of residues $ k $ and maximal ideal $ pW(k) $. Each element $ \omega \in W(k) $ can be uniquely represented as $$ \omega = \omega _{0} ^ \tau + p \omega _{1} ^ \tau + p ^{2} \omega _{2} ^ \tau + \dots , $$ where $ \omega _{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 \rightarrow W(k) $, splitting the mapping $$ W (k) \rightarrow W (k) / p \simeq k. $$ If $ k = \mathbf F _{p} $ is the prime field of $ p $ elements, $ W( \mathbf F _{p} ) $ is the ring of integral $ p $- adic numbers $ \mathbf Z _{p} $.


References

[1] E. Witt, "Zyklische Körper und Algebren der characteristik vom Grad . Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik " J. Reine Angew. Math. , 176 (1936) pp. 126–140 Zbl 0016.05101
[2] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[4] J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[5] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503
[6] J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique VII" Math. Ann. , 134 (1957) pp. 114–133


Comments

There is a generalization of the construction above which works for all primes $ p $ simultaneously, [a3]: a functor $ W : \ \mathbf{Ring} \rightarrow \mathbf{Ring} $ called the big Witt vector. Here, $ \mathbf{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,\dots \} $, let $ w _{n} (X) $ be the polynomial $$ w _{n} (X) = \sum _{d\mid n} dX _{d} ^{n/d} . $$ Then there is the following characterization theorem for the Witt vectors. There is a unique functor $ W : \ \mathbf{Ring} \rightarrow \mathbf{Ring} $ satisfying the following properties: 1) as a functor $ W: \ \mathbf{Ring} \rightarrow \mathop{\rm Set}\nolimits $, $ W (A) = \{ {(a _{1} ,\ a _{2} , \dots )} : {a _{i} \in A} \} $ and $ W ( \phi ) (a _{1} ,\ a _{2} , . . ) = ( \phi (a _{1} ) ,\ \phi (a _{2} ) ,\dots ) $ for any ring homomorphism $ \phi : \ A \rightarrow B $; 2) $ w _ {n , A} : \ W(A) \rightarrow A $, $ ( a _{1} ,\ a _{2} ,\dots ) \mapsto w _{n} (a _{1} ,\ a _{2} ,\dots ) $ is a functorial homomorphism of rings for every $ A $ and $ n \in \{ 1,\ 2,\dots \} $.


The functor $ W $ admits functorial ring endomorphisms $ \mathbf f _{n} : \ W \rightarrow W $, for every $ n \in \{ 1,\ 2,\dots \} $, that are uniquely characterized by $ w _{n} \mathbf f _{m} = w _{nm} $ for all $ n,\ m \in \{ 1,\ 2,\dots \} $. Finally, there is a functorial homomorphism $ \Delta : \ W(-) \rightarrow 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 _{1} \dots \Sigma _{n} ,\dots $; $ \Pi _{1} \dots \Pi _{n} ,\dots $; $ r _{1} \dots r _{n} ,\dots $ by the requirements $$ w _{n} ( \Sigma _{1} \dots \Sigma _{n} ) = w _{n} (X) + w _{n} (Y), $$ $$ w _{n} ( \Pi _{1} \dots \Pi _{n} ) = w _{n} (X) w _{n} (Y), $$ $$ w _{n} ( r _{1} \dots r _{n} ) = - w _{n} ( X) . $$ The $ \Sigma _{n} $ and $ \Pi _{n} $ are polynomials in $ X _{1} \dots X _{n} $; $ Y _{1} \dots Y _{n} $ and the $ r _{n} $ are polynomials in the $ X _{1} \dots X _{n} $ and they all have integer coefficients. $ W(A) $ is now defined as the set $ W(A) = \{ {\mathbf a = (a _{1} ,\ a _{2} ,\dots )} : {a _{i} \in A} \} $ with addition, multiplication and "minus" : $$ (a _{1} ,\ a _{2} ,\dots ) + (b _{1} ,\ b _{2} ,\dots ) = ( \Sigma _{1} ( \mathbf a ) ,\ \Sigma _{2} ( \mathbf a ) ,\dots ) $$ $$ (a _{1} ,\ a _{2} ,\dots ) (b _{1} ,\ b _{2} ,\dots ) = ( \Pi _{1} ( \mathbf a ) ,\ \Pi _{2} ( \mathbf a ) ,\dots ) - $$ $$ - (a _{1} ,\ a _{2} ,\dots ) = ( r _{1} ( \mathbf a ) ,\ r _{2} ( \mathbf a ) ,\dots ) . $$ The zero of $ W(A) $ is $ ( 0,\ 0 ,\dots ) $ and the unit element is $ ( 1,\ 0 ,\ 0 ,\dots ) $. 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(-) \rightarrow W(-) $, which are characterized by $$ w _{m} \mathbf V _{n} = \left \{ \begin{array}{ll} 0 & \textrm{ if } n \textrm{ does not divide } m, \\ nw _{m/n} & \textrm{ if } n \textrm{ divides } m. \\ \end{array} \right .$$ The $ \mathbf V _{m} $ are group endomorphisms of $ W(-) $ but not ring endomorphisms.

The ideals $ I _{n} = \{ ( 0 \dots 0,\ a _{n+1} ,\ a _{n+2} ,\dots ) \} \subset W(A) $ define a topology on $ W(A) $ making $ W(A) $ a separated complete topological ring.

For each $ A \in \mathbf{Ring} $, let $ \Lambda (A) $ be the Abelian group $ 1 + t A [[t]] $ under multiplication of power series; $$ \overline{E}\; : \ W(A) \rightarrow \Lambda (A), $$ $$ ( a _{1} ,\ a _{2} ,\dots ) \mapsto \prod _{i=1} ^ \infty (1- a _{i} t ^{i} ) , $$ defines a functional isomorphism of Abelian groups, and using the isomorphism $ \overline{E}\; $ there is a commutative ring structure on $ \Lambda (A) $. Using $ \overline{E}\; $ the Artin–Hasse exponential $ \Delta $ defines a functorial homomorphism of rings $$ W(A) \rightarrow \Lambda (W(A)) $$ making $ W(A) $ a functorial special $ \lambda $- ring. The Artin–Hasse exponential $ \Delta : \ W \rightarrow 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 $$ \mathbf f _{n} (1-at) = (1-a ^{n} t) , $$ $$ \mathbf V _{n} f(t) = f(t ^{n} ) , $$ and are completely determined by this (plus functoriality and additivity in the case of $ \mathbf f _{n} $).


For each supernatural number $ \mathbf n = \prod _{p} p ^ {\alpha _ p} $, $ \alpha _{p} \in \{ 0,\ 1,\ 2,\dots \} \cup \{ \infty \} $, one defines $ N ( \mathbf n ) = \{ {n \in \{ 1,\ 2,\dots \}} : {v _{p} (n) \leq \alpha _{p } \textrm{ for all "prime" numbers } p} \} $, where $ v _{p} $ is the $ p $- adic valuation of $ n $, i.e. the number of prime factors $ p $ in $ n $. Let $$ \mathfrak a _ {\mathbf n} (A) = $$ $$ = \{ {(a _{1} ,\ a _{2} ,\dots )} : { a _{d} = 0 \textrm{ for all } d \in N ( \mathbf n )} \} . $$ 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} (A) $, 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 \rightarrow W \circ W $ is compatible in a certain sense with the formation of these quotients, and using also the isomorphism $ \overline{E}\; $ one thus finds a mapping $$ \mathbf Z _{p} = W _ {p ^ \infty} ( \mathbf F _{p} ) \rightarrow \Lambda (W _ {p ^ \infty} ( \mathbf F _{p} ) ) = \Lambda ( \mathbf Z _{p} ) , $$ 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 [a1], [a2], [a3].

As an Abelian group $ W(A) $ is isomorphic to the group of curves $ {\mathcal C} ( \mathbf G _{m} ; \ A) $ of curves in the one-dimensional multiplicative formal group $ \mathbf G _{m} $. 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, [a3], [a4].

Let $ r _{n} (X,\ Y) $ be the sequence of polynomials with coefficients in $ \mathbf Z $ defined by $$ X ^{n} + Y ^{n} = \sum _{d\mid n} d r _{d} (X,\ Y) ^{n/d} . $$ The Cartier ring $ \mathop{\rm Cart}\nolimits (A) $ is the ring of all formal expressions $$ \tag{*} \sum _ {i,j \in \{ 1, 2,\dots \}} \mathbf V _{i} \langle a _{ij} \rangle \mathbf f _{j} $$ with the calculation rules $$ \langle a><b\rangle = \langle ab\rangle , \langle 1\rangle = \mathbf f _{1} = \mathbf V _{1} = \textrm{ unit element } 1 , $$ $$ \mathbf V _{n} \mathbf V _{m} = \mathbf V _{nm} , \mathbf f _{n} \mathbf f _{m} = \mathbf f _{nm} , $$ $$ \langle a\rangle \mathbf V _{m} = \mathbf V _{m} \langle a ^{m} \rangle , \mathbf f _{m} \langle a\rangle = \langle a ^{m} \rangle \mathbf f _{m} , $$ $$ \mathbf V _{m} \mathbf f _{n} = \mathbf f _{n} \mathbf V _{m} \textrm{ if } (n,\ m) = 1 , $$ $$ \mathbf f _{n} \mathbf V _{n} = 1 + \dots + 1 ( n \textrm{ summands } ) , $$ $$ \langle a+b\rangle = \sum _{n=1} ^ \infty \mathbf V _{n} \langle r _{n} ( a,\ b) \rangle \mathbf f _{n} . $$ Commutative formal groups over $ A $ are classified by certain modules over $ \mathop{\rm Cart}\nolimits (A) $. In case $ A $ is a $ \mathbf Z _{(p)} $- algebra, a simpler ring $ \mathop{\rm Cart}\nolimits _{p} (A) $ can be used for this purpose. It consists of all expressions (*) where now the $ i,\ j $ only run over the powers $ p ^{0} ,\ p ^{1} ,\ p ^{2} , . . . $ of the prime $ p $. The calculation rules are the analogous ones. In case $ k $ is a perfect field of characteristic $ p > 0 $ and $ \sigma $ denotes the Frobenius endomorphism of $ W(k) $( which in this case is given by $ \sigma ( a _{1} ,\ a _{2} , . . . ) = ( a _{1} ^{p} ,\ a _{2} ^{p} , . . ) $), then $ \mathop{\rm Cart}\nolimits _{p} (k) $ can be described as the ring of all expressions $$ x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} + \sum _{j=1} ^ \infty y _{j} \mathbf f ^{i} , $$ in two symbols $ \mathbf f $ and $ \mathbf V $ and with coefficients in $ W _ {p ^ \infty} (k) $, with the extra condition $ \mathop{\rm lim}\nolimits _ {i \rightarrow \infty} \ y _{i} = 0 $ and the calculation rules $$ \mathbf f x = \sigma (x) \mathbf f , \mathbf V x = \sigma ^{-1} (x) \mathbf V , $$ $$ \mathbf f \mathbf V = \mathbf V \mathbf f = p . $$ This ring, and also its subring of all expressions $$ x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} + \sum _{j=1} ^ {< \infty} y _{j} \mathbf f ^{j} , $$ is known as the Dieudonné ring $ D(k) $ and certain modules (called Dieudonné modules) over it classify unipotent commutative affine group schemes over $ k $, cf. [a5].

References

[a1] E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der -ten Potenzreste im Körper der -ten Einheitswurzeln" Abh. Math. Sem. Univ. Hamburg , 6 (1928) pp. 146–162
[a2] 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 MR73645
[a3] 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 MR0561822 Zbl 0437.13014
[a4] M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) MR506881
[a5] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503
How to Cite This Entry:
Witt vector. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Witt_vector&oldid=44326
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article