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].