# Formal group

An algebraic analogue of the concept of a local Lie group (cf. Lie group, local). The theory of formal groups has numerous applications in algebraic geometry, class field theory and cobordism theory.

A formal group over a field $k$ is a group object in the category of connected affine formal schemes over $k$ ( see [1], [4], [6], [7]). Let $H _{A}$ be the functor that associates with an algebra $B$ the set of algebra homomorphism $A \rightarrow B$ from some Noetherian commutative local $k$ - algebra $A$ with maximal ideal $m$ and field of residues $k$ , complete in the $m$ - adic topology, such that the homomorphisms map $m$ into the set $\mathop{\rm nil}\nolimits (B)$ of nilpotent elements of $B$ . Then a connected affine formal scheme is a covariant functor $H$ from the category of finite-dimensional commutative $k$ - algebras $B$ into the category of sets that is isomorphic to an $H _{A}$ . That $H$ is a group object means that there is a group structure given on all the sets $H (B)$ such that for every $k$ - algebra homomorphism $B _{1} \rightarrow B _{2}$ the corresponding mapping $H (B _{1} ) \rightarrow H (B _{2} )$ is a group homomorphism. If all the groups $H (B)$ are commutative, then the formal group $H$ is said to be commutative. Every connected group scheme $G$ over $k$ defines a formal group $G: \ B \rightarrow G (B)$ . Here one can take as $A$ the completion of the local ring of $G$ at the unit element.

If $A$ is the ring $k [[X _{1} \dots X _{2} ]]$ of formal power series in $n$ variables over $k$ , then $H$ is called an $n$ - dimensional formal Lie group. For a connected algebraic group $G$ over $k$ , $\widehat{G}$ is a formal Lie group. A formal Lie group $H$ is isomorphic, as a functor in the category of sets, to the functor $D ^{n} : \ B \rightarrow \mathop{\rm nil}\nolimits (B) ^{n}$ that associates with an algebra $B$ the $n$ - fold Cartesian product of its nil radical $\mathop{\rm nil}\nolimits (B)$ with itself. The group structure on the sets $H (B) = \mathop{\rm nil}\nolimits (B) ^{n}$ is given by a formal group law — a collection of $n$ formal power series in $2n$ variables $X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n}$ : $$F _{1} (X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n} ) \dots$$ $$F _{n} (X _{1} \dots X _{n} ,\ Y _{1} \dots Y _{n} ),$$ satisfying the following conditions:$$F _{i} (X,\ 0) = X _{i} , F _{i} (0,\ Y) = Y _{i} ,$$ $$F _{i} (X _{1} \dots X _{n} ,\ F _{1} (Y,\ Z) \dots F _{n} (Y,\ Z)) =$$ $$= F _{i} (F _{1} (X,\ Y) \dots F _{n} (X,\ Y),\ Z _{1} \dots Z _{n} ) .$$ Here $X = (X _{1} \dots X _{n} )$ , $Y = (Y _{1} \dots Y _{n} ),$ $Z = (Z _{1} \dots Z _{n} )$ , $0 = (0 \dots 0)$ . This group law on the sets $H (B) = \mathop{\rm nil}\nolimits (B) ^{n}$ is given by the formulas$$(x _{1} \dots x _{n} ) \circ (y _{1} \dots y _{n} ) = (z _{1} \dots z _{n} ),$$ where $z _{i} = F _{i} (x _{1} \dots x _{n} ,\ y _{1} \dots y _{n} )$ ; because $x$ and $y$ are nilpotent, all except a finite number of terms of the series are zero. Every formal group law gives group structures on $\mathop{\rm nil}\nolimits (B) ^{n}$ by means of

and converts the functor $D ^{n}$ into a formal Lie group. The concept of a formal group law, and thus of a formal Lie group, can be generalized to the case of arbitrary commutative ground rings (see [2], [5]). Sometimes by a formal group one means just a formal Lie group or even a formal group law.

Just as for local Lie groups (cf. Lie group, local) one can define the Lie algebra of a formal Lie group. Over fields $k$ of characteristic 0 the correspondence between a formal Lie group and its Lie algebra defines an equivalence of the respective categories. In characteristic $p > 0$ the situation is more complicated. Thus, over an algebraically closed field (for $p > 0$ ) there is a countable number of pairwise non-isomorphic one-dimensional commutative formal Lie groups [1], while all one-dimensional Lie algebras are isomorphic [3]. Over perfect fields of finite characteristic, commutative formal Lie groups are classified by means of Dieudonné modules (see [1], [6]).

The theory of formal groups over fields can be generalized to the case of arbitrary formal ground schemes [7].

#### References

 [1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 (1963) pp. 1–80 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90 MR157972 Zbl 0128.15603 [2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604 [3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803 [4] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 [5] M. Lazard, "Commutative formal groups" , Springer (1975) MR0393050 Zbl 0304.14027 [6] J.-M. Fontaine, "Groupes f04082068.png-divisibles sur les corps locaux" Astérique , 47–48 (1977) MR498610 [7] B. Mazur, J.T. Tate, "Canonical height pairings via biextensions" J.T. Tate (ed.) M. Artin (ed.) , Arithmetic and geometry , 1 , Birkhäuser (1983) pp. 195–237 MR0717595 Zbl 0574.14036

A universal formal group law (for $n$ - dimensional formal group laws) is an $n$ - dimensional formal group law $F _{u} ( X ,\ Y) \in L [ X ,\ Y ]$ , $X = ( X _{1} \dots X _{n} )$ , $Y = ( Y _{1} \dots Y _{n} )$ such that for every $n$ - dimensional formal group law $F ( X ,\ Y )$ over a ring $A$ there is a unique homomorphism of rings $\phi _{F} : \ L \rightarrow A$ such that $\phi _{F} ^{*} F _{u} ( X ,\ Y ) = F ( X ,\ Y )$ . Here $\phi _{F} ^{*} F _{u} ( X ,\ Y )$ denotes the result of applying $\phi _{F} ^{*}$ to the coefficients of the $n$ power series $F _{u} ( X ,\ Y )$ . Universal formal group laws exist and are unique in the sense that if $F _{u} ^ {\ \prime} ( X ,\ Y )$ over $L ^ \prime$ is another one, then there exists a ring isomorphism $\psi : \ L \rightarrow L ^ \prime$ such that $\psi ^{*} F _{u} ( X ,\ Y ) = F _{u} ^ {\ \prime} ( X ,\ Y )$ .

For commutative formal group laws explicit formulas are available for the construction of universal formal group laws, cf. [a3]. The underlying ring $L$ is a ring of polynomials in infinitely many indeterminates (Lazard's theorem).

A homomorphism of formal group laws $\alpha : \ F ( X ,\ Y ) \rightarrow G ( X ,\ Y )$ , $\mathop{\rm dim}\nolimits \ F = n$ , $\mathop{\rm dim}\nolimits \ G = m$ , is an $m$ - tuple of power series in $n$ - variables $\alpha ( Z _{1} \dots Z _{n} )$ , $\alpha _{i} (0) = 0$ , such that $\alpha ( F ( X ,\ Y ) ) = G ( \alpha (X) ,\ \alpha (Y))$ . The homomorphism is an isomorphism if there exists an inverse homomorphism $\beta$ such that $\alpha ( \beta (X) ) = X$ , and it is a strict isomorphism of formal group laws if $\alpha _{i} (Z) = Z _{i} +$ ( higher order terms).

Let $A$ be a ring of characteristic zero, i.e. the homomorphism of rings $\mathbf Z \rightarrow A$ which sends $1 \in \mathbf Z$ to the unit element in $A$ is injective. Then $A \rightarrow A \otimes _ {\mathbf Z} \mathbf Q$ is injective. Over $A \otimes _ {\mathbf Z} \mathbf Q$ all commutative formal group laws are strictly isomorphic and hence isomorphic to the additive formal group law$$G _{a} ( X ,\ Y ) = ( X _{1} + Y _{1} \dots X _{n} + Y _{n} ) .$$ It follows that for every commutative formal group law $F ( X ,\ Y )$ over $A$ there exists a unique $n$ - tuple of power series $f (X)$ , $f _{i} (X) = X _{i} + \dots$ with coefficients in $A \otimes _ {\mathbf Z} \mathbf Q$ such that$$F ( X ,\ Y ) = f ^ {\ -1} ( f (X) + f (Y) ) ,$$ where $f ^ {\ -1} (z)$ is the "inverse function" to $f (Z)$ , i.e. $f ^ {\ -1} ( f (Z) ) = Z$ . This $f (X)$ is called the logarithm of the group law $F ( X ,\ Y )$ .

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$$\mathop{\rm log}\nolimits \ F _ {\mathbf M \mathbf U} (X) = \sum _{i=1} ^ \infty i ^{-1} [ \mathbf C \mathbf P ^{i-1} ] X ^{i} .$$ 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 $\mathbf M \mathbf U ^{*} ( \mathop{\rm pt}\nolimits )$ . Cf. Cobordism for more details.

Let $F ( X ,\ Y )$ be an $n$ - dimensional group law over $A$ . A curve over $A$ in $F$ is an $n$ - tuple of power series $\gamma (T)$ in one variable such that $\gamma (0) = 0$ . Two curves can be added by $\gamma (T) + _{F} \delta (T) = F ( \gamma (T) ,\ \delta (T) )$ . The set of curves is given the natural power series topology and there results a commutative topological group ${\mathbf C} ( F ; \ A )$ . The group ${\mathbf C} ( F ; \ A )$ admits a number of operators $V _{n}$ , $F _{n}$ , $[a]$ , $a \in A$ , defined as follows:$$V _{n} \gamma (T) = \gamma ( T ^{n} ) ,$$ $$[a] \gamma (T) = \gamma ( a T ) ,$$ $$F _{n} \gamma (T) = \gamma ( \zeta _{n} T ^{1}/n ) + _{F} \dots + _{F} \gamma ( \zeta _{n} ^{n} T ^{1}/n ) = {\sum _ i=1 ^ n} {} ^{F} \gamma ( \zeta _{n} ^{i} T ^{1}/n ) ,$$ where $\zeta _{n}$ is a primitive $n$ - th root of unity. There are a number of relations between these operators and they combine to define a (non-commutative) ring $\mathop{\rm Cart}\nolimits (A)$ , which generalizes the Dieudonné ring, cf. Witt vector for the latter. Cartier's second and third theorems on formal group laws say that the $\mathop{\rm Cart}\nolimits (A)$ modules ${\mathbf C} ( F ; \ A )$ classify formal groups and they characterize which groups occur as ${\mathbf C} ( F ; \ A )$ ' 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 $W : \ \mathbf{Rinj} \rightarrow \mathbf{Rinj}$ be the functor of Witt vectors (cf. Witt vector). Let $G _{m} ( X ,\ Y ) = X + Y + X Y$ be the (one-dimensional) multiplicative formal group law over $A$ . Then $W (A) = {\mathbf C} ( G _{m} ; A )$ . Cartier's first theorem for formal group laws says that the functor $F \mapsto {\mathbf C} ( F ; \ A )$ is representable. More precisely, let $\widehat{W}$ be the (infinite-dimensional) formal group law given by the addition formulas of the Witt vectors and let $\gamma _{0} (T)$ be the curve $(T ,\ 0 ,\ 0 ,\dots )$ . Then for every formal group law $F ( X ,\ Y )$ and curve $\gamma (T) \in {\mathbf C} ( F ; \ A )$ there is unique homomorphism of formal group laws $\alpha _ \gamma : \ \widehat{W} \rightarrow F$ such that $\alpha _ \gamma ( \gamma _{0} (T) ) = \gamma (T)$ .

There exists a Pontryagin-type duality between commutative formal groups and commutative affine (algebraic) groups over a field $k$ , 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 $A$ be a discrete valuation ring with finite residue field $k$ and maximal ideal $\mathfrak m = ( \pi )$ . The Lubin–Tate formal group law associated to $( A ,\ \pi )$ is defined by the logarithm$$f _ \pi (X) = X + \pi ^{-1} X ^{q} + \pi ^{-2} X ^ {q ^{2}} + \dots .$$ Then $F _ \pi ( X ,\ Y ) = f _ \pi ^ {\ -1} ( f _ \pi (X) + f _ \pi (Y) )$ has its coefficients in $A$ . These formal group laws are in a sense formal $p$ - 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 $K$ , the quotient field of $A$ . Indeed, let $F _ \pi ( \bar{\mathfrak m} )$ be the set $\bar{\mathfrak m}$ with the addition $a + b = F _ \pi ( a ,\ b )$ . Here $\bar{\mathfrak m}$ is the maximal ideal of the ring of integers of an algebraic closure of $K$ . Then a maximal Abelian totally ramified extension of $K$ is generated by the torsion elements of $F ( \bar{\mathfrak m} )$ ; 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: $p$ - divisible groups; cf. $p$ - 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 $F ( X ,\ Y )$ be a one-dimensional formal group law. Define inductively $[n] (X) = F ( X ,\ [ n - 1 ] (X) )$ , $n \geq 2$ , $[1] (X) = X$ . Let $F$ be defined over a field $k$ of characteristic $p > 0$ . Then $[p] (X)$ is necessarily of the form $X ^ {p ^{h}} +$ ( higher degree terms) or is equal to zero. The positive integer $h$ is called the height of $F$ ; if $[p] (X) = 0$ , the height of $F$ is taken to be $\infty$ . Over an algebraically closed field of characteristic $p > 0$ the one-dimensional formal group laws are classified by their heights, and all heights $1 ,\ 2 \dots \infty$ occur.

Let $F ( X ,\ Y )$ be a one-dimensional formal group law over a ring $A$ in which every prime number except $p$ is invertible, e.g. $A$ is the ring of integers of a local field of residue characteristic $p$ or $A$ is a field of characteristic $p$ . Assume for the moment that $A$ is of characteristic zero and let$$f (X) = X + a _{2} X ^{2} + a _{3} X ^{3} + \dots$$ be the logarithm of $F ( X ,\ Y )$ . Then $F ( X ,\ Y )$ is strictly isomorphic over $A$ to the formal group law $F _{(p)} ( X ,\ Y )$ whose logarithm is equal to$$f _{(p)} (X) = X + a _{p} X ^{p} + a _ {p ^{2}} X ^ {p ^{2}} + \dots .$$ The result extends to the case that $A$ is not of characteristic zero and to more-dimensional commutative formal group laws. $F _{(p)} ( X ,\ Y)$ is called the $p$ - typification of $F ( X ,\ Y )$ .

#### References

 [a1] J.T. Tate, "-divisible groups" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183 MR0231827 Zbl 0157.27601 [a2] J.-P. Serre, "Groupes -divisible (d' après J. Tate)" Sem. Bourbaki , 19, Exp. 318 (1966–1967) MR1610452 MR0393040 [a3] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020 [a4] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson & North-Holland (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 [a5] J. Lubin, J. Tate, "Formal complex multiplication in local fields" Ann. of Math. , 81 (1965) pp. 380–387 MR0172878 Zbl 0128.26501
How to Cite This Entry:
Formal group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_group&oldid=44249
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article