An element of an algebraic construct, first proposed by E. Witt [1] in 1936 in the context of the description of unramified extensions of 
-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 
be an associative, commutative ring with unit element. Witt vectors with components in 
are infinite sequences 
, 
, which are added and multiplied in accordance with the following rules:
where 
, 
are polynomials in the variables 
, 
with integer coefficients, uniquely defined by the conditions
where
are polynomials, 
and 
is a prime number. In particular,
The Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by 
. For any natural number 
there also exists a definition of the ring 
of truncated Witt vectors of length 
. The elements of this ring are finite tuples 
, 
, with the addition and multiplication operations described above. The canonical mappings
are homomorphisms. The rule 
(or 
) 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 
(or 
) on which the structure of a ring object has been defined. The spectrum 
(or 
) is known as a Witt scheme (or a truncated Witt scheme) and is a ring scheme [3].
Each element 
defines a Witt vector
called the Teichmüller representative of the element 
. If 
is a perfect field of characteristic 
, 
is a complete discrete valuation ring of zero characteristic with field of residues 
and maximal ideal 
. Each element 
can be uniquely represented as
where 
. Conversely, each such ring 
with field of residues 
is canonically isomorphic to the ring 
. The Teichmüller representation makes it possible to construct a canonical multiplicative homomorphism 
, splitting the mapping
If 
is the prime field of 
elements, 
is the ring of integral 
-adic numbers 
.
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 |
| [2] | S. Lang, "Algebra" , Addison-Wesley (1974) |
| [3] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) |
| [4] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) |
| [5] | M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) |
| [6] | J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique  VII" Math. Ann. , 134 (1957) pp. 114–133 |
There is a generalization of the construction above which works for all primes 
simultaneously, [a3]: a functor 
called the big Witt vector. Here, 
is the category of commutative, associative rings with unit element. The functor described above, of Witt vectors of infinite length associated to the prime 
, is a quotient of 
which can be conveniently denoted by 
.
For each 
, let 
be the polynomial
Then there is the following characterization theorem for the Witt vectors. There is a unique functor 
satisfying the following properties: 1) as a functor 
, 
and 
for any ring homomorphism 
; 2) 
, 
is a functorial homomorphism of rings for every 
and 
.
The functor 
admits functorial ring endomorphisms 
, for every 
, that are uniquely characterized by 
for all 
. Finally, there is a functorial homomorphism 
that is uniquely characterized by the property 
for all 
, 
.
To construct 
, define polynomials 
; 
; 
by the requirements
The 
and 
are polynomials in 
; 
and the 
are polynomials in the 
and they all have integer coefficients. 
is now defined as the set 
with addition, multiplication and "minus" :
The zero of 
is 
and the unit element is 
. The Frobenius endomorphisms 
and the Artin–Hasse exponential 
are constructed by means of similar considerations, i.e. they are also given by certain universal polynomials. In addition there are the Verschiebung morphisms 
, which are characterized by
The 
are group endomorphisms of 
but not ring endomorphisms.
The ideals 
define a topology on 
making 
a separated complete topological ring.
For each 
, let 
be the Abelian group 
under multiplication of power series;
defines a functional isomorphism of Abelian groups, and using the isomorphism 
there is a commutative ring structure on 
. Using 
the Artin–Hasse exponential 
defines a functorial homomorphism of rings
making 
a functorial special 
-ring. The Artin–Hasse exponential 
defines a cotriple structure on 
and the co-algebras for this co-triple are precisely the special 
-rings (cf. also Category and Triple).
On 
the Frobenius and Verschiebung endomorphisms satisfy
and are completely determined by this (plus functoriality and additivity in the case of 
).
For each supernatural number 
, 
, one defines 
, where 
is the 
-adic valuation of 
, i.e. the number of prime factors 
in 
. Let
Then 
is an ideal in 
and for each supernatural 
a corresponding ring of Witt vectors is defined by
In particular, one thus finds 
, the ring of infinite-length Witt vectors for the prime 
, discussed in the main article above, as a quotient of the ring of big Witt vectors 
.
The Artin–Hasse exponential 
is compatible in a certain sense with the formation of these quotients, and using also the isomorphism 
one thus finds a mapping
where 
denotes the 
-adic integers and 
the field of 
elements, which can be identified with the classical morphism defined by Artin and Hasse [a1], [a2], [a3].
As an Abelian group 
is isomorphic to the group of curves 
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, [a3], [a4].
Let 
be the sequence of polynomials with coefficients in 
defined by
The Cartier ring 
is the ring of all formal expressions
 | (*) |
with the calculation rules
Commutative formal groups over 
are classified by certain modules over 
. In case 
is a 
-algebra, a simpler ring 
can be used for this purpose. It consists of all expressions (*) where now the 
only run over the powers 
of the prime 
. The calculation rules are the analogous ones. In case 
is a perfect field of characteristic 
and 
denotes the Frobenius endomorphism of 
(which in this case is given by 
), then 
can be described as the ring of all expressions
in two symbols 
and 
and with coefficients in 
, with the extra condition 
and the calculation rules
This ring, and also its subring of all expressions
is known as the Dieudonné ring 
and certain modules (called Dieudonné modules) over it classify unipotent commutative affine group schemes over 
, 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 |
| [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 |
| [a4] | M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) |
| [a5] | M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) |
