A formal group is a group object internal to infinitesimal spaces. More general than Lie algebras, which are group objects in first order infinitesimal spaces, formal groups may be of arbitrary infinitesimal order. They sit between Lie algebras and finite Lie groups or algebraic groups.
Specifically, fixing a formal coordinate chart, then the product operation of a formal group is entirely expressed as a formal power series in two variables, satisfying conditions. This is called a formal group law, a concept that goes back to Bochner and Lazard.
Commutative formal group laws of dimension 1 notably appear in algebraic topology (originating in work by Novikov, Buchstaber and Quillen, see Adams 74, part II), where they express the behaviour of complex oriented cohomology theories evaluated on infinite complex projective space (i.e. on the classifying space ). In particular complex cobordism cohomology theory in this context gives the universal formal group law represented by the Lazard ring. The height of formal groups induces a filtering on complex oriented cohomology theories called the chromatic filtration.
More recently Morel and Marc Levine consider the algebraic cobordism of smooth schemes in algebraic geometry. Formal groups are also useful in local class field theory; they can be used to explicitly construct the local Artin map according to Lubin and Tate.
the canonical morphism
A homomorphism of adic rings is a ring homomorphism that is also a continuous function (hence a function that preserves the filtering ). This gives a category and a subcategory of commutative adic rings.
Similarly, for any fixed commutative ring, then adic rings under are adic -algebras. We write and for the corresponding categories.
is an adic ring (def. 1).
There is a fully faithful functor
i.e. for two adic rings, then there is a natural isomorphism
Hence this is a morphism
in satisfying unitality, associativity.
This is a commutative formal group law if it is an abelian group object, hence if it in addition satisfies the corresponding commutativity condition.
This is equivalently a set of power series of variables such that (in notation , , )
(the image under in of the element ) such that
The first condition means equivalently that
Hence is necessarily of the form
The existence of inverses is no extra condition: by induction on the index one finds that there exists a unique
Hence 1-dimensional formal group laws over are equivalently monoids in on .
Much more general are formal group schemes from (Grothendieck)
Formal group schemes are simply the group objects in a category of formal schemes; however usually only the case of the formal spectra of complete -algebras is considered; this category is equivalent to the category of complete cocommutative -Hopf algebras.
The quotient moduli stack of formal group over the rational numbers is isomorphic to , the delooping of the multiplicative group (over ). This means that in characteristic 0 every formal group is determined, up to unique isomorphism, by its Lie algebra.
For instance (Lurie 10, lecture 12, corollary 3).
It is immediate that there exists a ring carrying a universal formal group law. For observe that for an element in a formal power series algebra, then the condition that it defines a formal group law is equivalently a sequence of polynomial equations on the coefficients . For instance the commutativity condition means that
and the unitality constraint means that
Similarly associativity is equivalently a condition on combinations of triple products of the coefficients. It is not necessary to even write this out, the important point is only that it is some polynomial equation.
This allows to make the following definition
quotiented by the relations
the obvious associativity relation
for all .
The following is immediate from the definition:
If the formal group law has coefficients , then in order that , i.e. that
it must be that is given by
where are the generators of the Lazard ring. Hence it only remains to see that this indeed constitutes a ring homomorphism. But this is guaranteed by the vary choice of relations imposed in the definition of the Lazard ring.
What is however highly nontrivial is this statement:
in countably many generators in degree .
The Lazard theorem 1 first of all implies, via prop. 3, that there exists an abundance of 1-dimensional formal group laws: given any ring then every choice of elements defines a formal group law. (On the other hand, it is nontrivial to say which formal group law that is.)
Deeper is the fact expressed by the Milnor-Quillen theorem on MU: the Lazard ring in its polynomial incarnation of prop. 1 is canonically identieif with the graded commutative ring of stable homotopy groups of the universal complex Thom spectrum MU. Moreover:
under forming stable homtopy groups every ring spectrum homomorphism induces a ring homomorphism
and hence, by the universality of , a formal group law over .
This is the formal group law given by the above complex orientation.
Formal geometry is closely related also to the rigid analytic geometry.
(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||germ of a space||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
Shigkaki Tôgô, Note of formal Lie groups , American Journal of Mathematics, Vol. 81, No. 3, Jul., 1959 (JSTOR)
A. Fröhlich, Formal group, Lecture Notes in Mathematics Volume 74, Springer (1968)
Stanley Kochmann, section 4.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Michiel Hazewinkel, Formal Groups and Applications, projecteuclid
Jean Dieudonné, Introduction to the theory of formal groups, Marcel Dekker, New York 1973.
Quillen's theorem on MU is due to
A basic introduction is in