A
universal formal group law
(for
-dimensional
formal group laws) is an
-dimensional
formal group law
,
,
such that for every
-dimensional
formal group law
over a ring
there is a unique homomorphism of rings
such that
.
Here
denotes the result of applying
to the coefficients of the
power series
.
Universal formal group laws exist and are unique in the sense that if
over
is another one, then there exists a ring isomorphism
such that
.
For commutative formal group laws explicit formulas are available
for the construction of universal formal group laws, cf.
[a3].
The underlying ring
is a ring of polynomials in infinitely many indeterminates
(Lazard's theorem).
A
homomorphism of formal group laws
,
,
,
is an
-tuple
of power series in
-variables
,
,
such that
.
The homomorphism is an
isomorphism
if there exists an inverse homomorphism
such that
,
and it is a
strict isomorphism
of formal group laws if
(higher
order terms).
Let
be a ring of characteristic zero, i.e. the homomorphism of rings
which sends
to the unit element in
is injective. Then
is injective. Over
all commutative formal group laws are strictly isomorphic and hence isomorphic to the
additive formal group law
It follows that for every commutative formal group law
over
there exists a unique
-tuple
of power series
,
with coefficients in
such that
where
is the
"inverse function"
to
,
i.e.
.
This
is called the
logarithm
of the group law
.
The formal group law of complex
cobordism
is a universal one-dimensional formal group law
(Quillen's theorem)
and its logarithm is given by
Mishchenko's formula
Combined with the explicit construction of a one-dimensional universal group law
these facts yield useful information on the generators of the complex cobordism ring
.
Cf.
Cobordism
for more details.
Let
be an
-dimensional
group law over
.
A
curve
over
in
is an
-tuple
of power series
in one variable such that
.
Two curves can be added by
.
The set of curves is given the natural power
series topology and there results a commutative topological group
.
The group
admits a number of operators
,
,
,
,
defined as follows:
where
is a primitive
-th
root of unity. There are a number of relations
between these operators and they combine to define a (non-commutative) ring
,
which generalizes the
Dieudonné ring,
cf.
Witt vector
for the latter.
Cartier's second and third theorems on formal group laws
say that the
modules
classify formal groups and they characterize which groups occur as
's.
This is the covariant classification of commutative formal groups in
contrast with the earlier contravariant classification of commutative
formal groups over perfect fields by Dieudonné modules.
Let
be the functor of Witt vectors (cf.
Witt vector).
Let
be the (one-dimensional)
multiplicative formal group law
over
.
Then
.
Cartier's first theorem for formal group laws
says that the functor
is representable. More precisely, let
be the (infinite-dimensional) formal group law given by the
addition formulas of the Witt vectors and let
be the curve
.
Then for every formal group law
and curve
there is unique homomorphism of formal group laws
such that
.
There exists a Pontryagin-type duality between commutative formal groups
and commutative affine (algebraic) groups over a field
,
called
Cartier duality.
Cf.
[a3],
[a4]
for more details. Correspondingly, Dieudonné modules are also important
in the classification of commutative affine (algebraic) groups. Essentially, Cartier
duality comes from the
"duality"
between algebras and co-algebras; cf.
Co-algebra.
Let
be a discrete valuation ring with finite residue field
and maximal ideal
.
The
Lubin–Tate formal group law
associated to
is defined by the logarithm
Then
has its coefficients in
.
These formal group laws are in a sense formal
-adic
analogues of elliptic curves with complex multiplication in that they
have maximally large endomorphism rings. They are also analogues in the
role they play vis à vis the class field theory of
,
the quotient field of
.
Indeed, let
be the set
with the addition
.
Here
is the maximal ideal of the ring of integers of an algebraic closure of
.
Then a maximal Abelian totally ramified extension of
is generated by the torsion elements of
;
cf.
[a3],
[a5]
for more details.
Formal groups also are an important tool in algebraic geometry, especially
in the theory of Abelian varieties. This holds even more so for a generalization:
-divisible
groups; cf.
-divisible group.
Lazard's theorem on one-dimensional formal group laws
says that all one-dimensional formal group laws
over a ring without nilpotents are commutative.
Let
be a one-dimensional formal group law. Define inductively
,
,
.
Let
be defined over a field
of characteristic
.
Then
is necessarily of the form
(higher degree terms) or is equal to zero. The positive integer
is called the
height
of
;
if
,
the height of
is taken to be
.
Over an algebraically closed field of characteristic
the one-dimensional formal group laws are classified by their heights, and all heights
occur.
Let
be a one-dimensional formal group law over a ring
in which every prime number except
is invertible, e.g.
is the ring of integers of a local field of residue characteristic
or
is a field of characteristic
.
Assume for the moment that
is of characteristic zero and let
be the logarithm of
.
Then
is strictly isomorphic over
to the formal group law
whose logarithm is equal to
The result extends to the case that
is not of characteristic zero and to more-dimensional commutative formal group laws.
is called the
-typification
of
.