What is a formal group?

Índice Show

Formal group laws
Formal group schemes
Formal groups over an operad
In characteristic 0
Universal 1d commutative formal group law
1-Dimensional formal groups

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.

Since infinitesimal spaces are typically modeled as formal duals to algebras, formal groups are typically conceived as group objects in formal duals to formal power series algebras.

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 BU(1)≃ℂP ∞B U(1) \simeq \mathbb{C}P^\infty). 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.

Formal group laws

For RR a commutative ring and n∈ℕn \in \mathbb{N} then the formal power series ring

R[[x 1,x 2,⋯,x n]] R[ [ x_1, x_2, \cdots, x_n ] ]

in nn variables with coefficients in RR and equipped with the ideal

I=(x 1,⋯,x n) I = (x_1, \cdots , x_n)

is an adic ring (def. ).

There is a fully faithful functor

AdicRing↪ProRing AdicRing \hookrightarrow ProRing

from adic rings (def. ) to pro-rings, given by

(A,I)↦((A/I •)), (A,I) \mapsto ( (A/I^{\bullet})) \,,

i.e. for A,B∈AdicRingA,B \in AdicRing two adic rings, then there is a natural isomorphism

Hom AdicRing(A,B)≃lim⟵ n 2lim⟶ n 1Hom Ring(A/I n 1,B/I n 2). Hom_{AdicRing}(A,B) \simeq \underset{\longleftarrow}{\lim}_{n_2} \underset{\longrightarrow}{\lim}_{n_1} Hom_{Ring}(A/I^{n_1},B/I^{n_2}) \,.

For R∈CRingR \in CRing a commutative ring and for n∈ℕn \in \mathbb{N}, a formal group law of dimension nn over RR is the structure of a group object in the category AdicRCAlg opAdic R CAlg^{op} from def. on the object R[[x 1,⋯,x n]]R [ [x_1, \cdots ,x_n] ] from example .

Hence this is a morphism

μ:R[[x 1,⋯,x n]]⟶R[[x 1,⋯,x n,y 1,⋯,y n]] \mu \;\colon\; R[ [ x_1, \cdots, x_n ] ] \longrightarrow R [ [ x_1, \cdots, x_n, \, y_1, \cdots, y_n ] ]

in AdicRCAlgAdic R CAlg 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 nn power series F iF_i of 2n2n variables x 1,…,x n,y 1,…,y nx_1,\ldots,x_n,y_1,\ldots,y_n such that (in notation x=(x 1,…,x n)x=(x_1,\ldots,x_n), y=(y 1,…,y n)y=(y_1,\ldots,y_n), F(x,y)=(F 1(x,y),…,F n(x,y))F(x,y) = (F_1(x,y),\ldots,F_n(x,y)))

F(x,F(y,z))=F(F(x,y),z) F(x,F(y,z))=F(F(x,y),z)

F i(x,y)=x i+y i+higherorderterms F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms

A 1-dimensional commutative formal group law according to def. is equivalently a formal power series

μ(x,y)=∑i,j≥0a i,jx iy j \mu(x,y) = \underset{i,j \geq 0}{\sum} a_{i,j} x^i y^j

(the image under μ\mu in R[[x,y]]R[ [ x,y ] ] of the element t∈R[[t]]t \in R [ [ t ] ]) such that

(unitality)
(associativity)

μ(x,μ(y,z))=μ(μ(x,y),z); \mu(x,\mu(y,z)) = \mu(\mu(x,y),z) \,;
(commutativity)

μ(x,y)=μ(y,x). \mu(x,y) = \mu(y,x) \,.

The first condition means equivalently that

a i,0={1 ifi=1 0 otherwise,a 0,i={1 ifi=1 0 otherwise. a_{i,0} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \;\;\;\;\,, \;\;\;\;\; a_{0,i} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \,.

Hence μ\mu is necessarily of the form

μ(x,y)=x+y+∑i,j≥1a i,jx iy j. \mu(x,y) \;=\; x + y + \underset{i,j \geq 1}{\sum} a_{i,j} x^i y^j \,.

The existence of inverses is no extra condition: by induction on the index ii one finds that there exists a unique

ι(x)=∑i≥1ι(x) ix i \iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i

such that

μ(x,ι(x))=0,μ(ι(x),x)=0. \mu(x,\iota(x)) = 0 \;\;\;\,, \;\;\; \mu(\iota(x),x) = 0 \,.

Hence 1-dimensional formal group laws over RR are equivalently monoids in AdicRCAlg opAdic R CAlg^{op} on R[[x]]R[ [ x ] ].

Any power series of the form f(x)=x+a 2x 2+a 3x 3+…f(x) = x + a_2 x^2 + a_3 x^3 + \ldots in R[[x]]R[ [x] ] has a functional or compositional inverse f −1(x)f^{-1}(x) in the monoid xR[[x]]x R[ [x] ] under composition. Thus we may define a 1-dimensional formal group law by the formula μ(x,y)=f −1(f(x)+f(y))\mu(x, y) = f^{-1}(f(x) + f(y)). That this is in some sense the typical way that 1-dimensional formal group laws arise is the content of Lazard's theorem.

Formal group schemes

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 kk-algebras is considered; this category is equivalent to the category of complete cocommutative kk-Hopf algebras.

Formal groups over an operad

For a generalization over operads see (Fresse).

Properties

In characteristic 0

For instance (Lurie 10, lecture 12, corollary 3).

Universal 1d commutative formal group law

It is immediate that there exists a ring carrying a universal formal group law. For observe that for ∑i,ja i,jx 1 ix 1 j\underset{i,j}{\sum} a_{i,j} x_1^i x_1^j 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 a ka_k. For instance the commutativity condition means that

a i,j=a j,i a_{i,j} = a_{j,i}

and the unitality constraint means that

a i0={1 ifi=1 0 otherwise. a_{i 0} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \,.

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

The Lazard ring is the graded commutative ring generated by elements a ija_{i j} in degree 2(i+j−1)2(i+j-1) with i,j∈ℕi,j \in \mathbb{N}

L=ℤ[a ij]/(relations1,2,3below) L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)

quotiented by the relations

a ij=a jia_{i j} = a_{j i}
a 10=a 01=1a_{10} = a_{01} = 1; ∀i≠1:a i0=0\forall i \neq 1: a_{i 0} = 0
the obvious associativity relation

for all i,j,ki,j,k.

The universal 1-dimensional commutative formal group law is the formal power series with coefficients in the Lazard ring given by

ℓ(x,y)≔∑ i,ja ijx iy j∈L[[x,y]]. \ell(x,y) \coloneqq \sum_{i,j} a_{i j} x^i y^j \in L[ [ x , y ] ] \,.

The following is immediate from the definition:

For every ring RR and 1-dimensional commutative formal group law μ\mu over RR (example ), there exists a unique ring homomorphism

f:L⟶R f \;\colon\; L \longrightarrow R

from the Lazard ring (def. ) to RR, such that it takes the universal formal group law ℓ\ell to μ\mu

f *ℓ=μ. f_\ast \ell = \mu \,.

If the formal group law μ\mu has coefficients {c i,j}\{c_{i,j}\}, then in order that f *ℓ=μf_\ast \ell = \mu, i.e. that

∑i,jf(a i,j)x iy j=∑i,jc i,jx iy j \underset{i,j}{\sum} f(a_{i,j}) x^i y^j = \underset{i,j}{\sum} c_{i,j} x^i y^j

it must be that ff is given by

f(a i,j)=c i,j f(a_{i,j}) = c_{i,j}

where a i,ja_{i,j} 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:

Examples

Formal geometry is closely related also to the rigid analytic geometry.

(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).

References

General

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)
Alexander Grothendieck et al. SGA III, vol. 1, Expose VIIB (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES (part B) 474-560
Frank Adams, Part II.1 of Stable homotopy and generalised homology, 1974
Stanley Kochmann, section 4.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Benoit Fresse, Lie theory of formal groups over an operad, J. Alg. 202, 455–511, 1998, doi
Michiel Hazewinkel, Formal Groups and Applications, projecteuclid
Michel Demazure, lectures on p-divisible groups
Jean Dieudonné, Introduction to the theory of formal groups, Marcel Dekker, New York 1973.

1-Dimensional formal groups

A basic introduction is in

Carl Erickson, One-dimensional formal groups (pdf)

Specifically formal group laws of elliptic curves:

Antonia W. Bluher, Formal groups, elliptic curves, and some theorems of Couveignes, in: J.P. Buhler (eds.) Algorithmic Number Theory ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 (arXiv:math/9708215, doi:10.1007/BFb0054887)
Stefan Friedl, An elementary proof of the group law for elliptic curves (arXiv:1710.00214)

Quillen's theorem on MU is due to