formal group


\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Formal geometry

Group Theory



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)P 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


An (commutative) adic ring is a (commutative) topological ring AA and an ideal IAI \subset A such that

  1. the topology on AA is the II-adic topology;

  2. the canonical morphism

    Alim n(A/I n) A \longrightarrow \underset{\longleftarrow}{\lim}_n (A/I^n)

    to the limit over quotient rings by powers of the ideal is an isomorphism.

A homomorphism of adic rings is a ring homomorphism that is also a continuous function (hence a function that preserves the filtering AA/I 2A/IA \supset \cdots \supset A/I^2 \supset A/I ). This gives a category AdicRingAdicRing and a subcategory AdicCRingAdicCRing of commutative adic rings.

The opposite category of AdicRingAdicRing (on Noetherian rings) is that of affine formal schemes.

Similarly, for RR any fixed commutative ring, then adic rings under RR are adic RR-algebras. We write AdicAAlgAdic A Alg and AdicACAlgAdic A CAlg for the corresponding categories.


For RR a commutative ring and 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. 1).


There is a fully faithful functor

AdicRingProRing AdicRing \hookrightarrow ProRing

from adic rings (def. 1) to pro-rings, given by

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

i.e. for A,BAdicRingA,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 RCRingR \in CRing a commutative ring and for 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. 1 on the object R[[x 1,,x n]]R [ [x_1, \cdots ,x_n] ] from example 1.

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. 2 is equivalently a formal power series

μ(x,y)=i,j0a 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 tR[[t]]t \in R [ [ t ] ]) such that

  1. (unitality)

    μ(x,0)=x \mu(x,0) = x
  2. (associativity)

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

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

The first condition means equivalently that

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

Hence μ\mu is necessarily of the form

μ(x,y)=x+y+i,j1a 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)=i1ι(x) ix i \iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i

such that

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

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

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).


In characteristic 0


The quotient moduli stack FG×Spec\mathcal{M}_{FG} \times Spec \mathbb{Q} of formal group over the rational numbers is isomorphic to B𝔾 m\mathbf{B}\mathbb{G}_m, the delooping of the multiplicative group (over SpecSpec \mathbb{Q}). 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).

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+j1)2(i+j-1) with i,ji,j \in \mathbb{N}

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

quotiented by the relations

  1. a ij=a jia_{i j} = a_{j i}

  2. a 10=a 01=1a_{10} = a_{01} = 1; i1:a i0=0\forall i \neq 1: a_{i 0} = 0

  3. 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

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

The grading is chosen with regards to the formal group laws arising from complex oriented cohomology theories (prop.) where the variable xx naturally has degree -2. This way

deg(a ijx iy j)=deg(a i,j)+ideg(x)+jdeg(y)=2. deg(a_{i j} x^i y^j) = deg(a_i,j) + i deg(x) + j deg(y) = -2 \,.

The following is immediate from the definition:


For every ring RR and 1-dimensional commutative formal group law μ\mu over RR (example 2), there exists a unique ring homomorphism

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

from the Lazard ring (def. 3) to RR, such that it takes the universal formal group law \ell to μ\mu

f *=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 *=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:


(Lazard's theorem)

The Lazard ring LL (def. 3) is isomorphic to a polynomial ring

L[t 1,t 2,] L \simeq \mathbb{Z}[ t_1, t_2, \cdots ]

in countably many generators t it_i in degree 2i2 i.


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 RR then every choice of elements {t iR}\{t_i \in R\} 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 π (MU)\pi_\bullet(M U) of stable homotopy groups of the universal complex Thom spectrum MU. Moreover:

  1. MU carries a universal complex orientation in that for EE any homotopy commutative ring spectrum then homotopy classes of homotopy ring homomorphisms MUEM U \to E are in bijection to complex orientations on EE;

  2. every complex orientation on EE induced a 1-dimensional commutative formal group law (prop.)

  3. under forming stable homtopy groups every ring spectrum homomorphism MUEM U \to E induces a ring homomorphism

    Lπ (MU)π (E) L \simeq \pi_\bullet(M U) \longrightarrow \pi_\bullet(E)

    and hence, by the universality of LL, a formal group law over π (E)\pi_\bullet(E).

This is the formal group law given by the above complex orientation.

Hence the universal group law over the Lazard ring is a kind of decategorification of the universal complex orientation on MU.


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

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization



Quillen's theorem on MU is due to

1-Dimensional formal groups

A basic introduction is in

  • Carl Erickson, One-dimensional formal groups (pdf)

An elementary discussion of associativity of the formal group law of elliptic curves is in

  • Stefan Friedl, An elementary proof of the group law for elliptic curves Stefan Friedl (arXiv:1710.00214)

See also

  • Takeshi Torii, One dimensional formal group laws of height NN and N1N-1, PhD thesis 2001 (pdf)

  • Takeshi Torii, On Degeneration of One-Dimensional Formal Group Laws and Applications to Stable Homotopy Theory, American Journal of Mathematics Vol. 125, No. 5 (Oct., 2003), pp. 1037-1077 (JSTOR)

  • Stefan Schwede, Formal groups and stable homotopy of commutative rings, Geom. Topol. 8 (2004) 335-412 (arXiv:math/0402372)

The moduli stack of formal groups and its incarnation as a Hopf algebroid:

Revised on November 2, 2017 05:08:24 by Urs Schreiber (