User:Maximilian Janisch/latexlist/Algebraic Groups/Formal group

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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 $4$ with maximal ideal $m$ and field of residues $k$, complete in the $m$-adic topology, such that the homomorphisms map $m$ into the set $\operatorname { nil } ( B )$ 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 $k$ over $k$ defines a formal group $G : B \rightarrow G ( B )$. Here one can take as $4$ the completion of the local ring of $k$ at the unit element.

If $4$ is the ring $k [ [ X _ { 1 } , \ldots , X _ { 2 } ] ]$ of formal power series in $12$ variables over $k$, then $H$ is called an $12$-dimensional formal Lie group. For a connected algebraic group $k$ over $k$, $\vec { C }$ is a formal Lie group. A formal Lie group $H$ is isomorphic, as a functor in the category of sets, to the functor $D ^ { X } : B \rightarrow \operatorname { nil } ( B ) ^ { n }$ that associates with an algebra $B$ the $12$-fold Cartesian product of its nil radical $\operatorname { nil } ( B )$ with itself. The group structure on the sets $H ( B ) = \operatorname { nil } ( B ) ^ { n }$ is given by a formal group law — a collection of $12$ formal power series in $2 n$ variables $X _ { 1 } , \ldots , X _ { X } , Y _ { 1 } , \ldots , Y _ { X }$:

\begin{equation} F _ { 1 } ( X _ { 1 } , \ldots , X _ { x } , Y _ { 1 } , \ldots , Y _ { n } ) \end{equation}

\begin{equation} F _ { n } ( X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n } ) \end{equation}

satisfying the following conditions:

\begin{equation} F _ { i } ( X , 0 ) = X _ { i } , \quad F _ { i } ( 0 , Y ) = Y _ { i } \end{equation}

\begin{equation} F _ { i } ( X _ { 1 } , \ldots , X _ { n } , F _ { 1 } ( Y , Z ) , \ldots , F _ { n } ( Y , Z ) ) = \end{equation}

\begin{equation} = F _ { i } ( F _ { 1 } ( X , Y ) , \ldots , F _ { n } ( X , Y ) , Z _ { 1 } , \ldots , Z _ { n } ) \end{equation}

Here $X = ( X _ { 1 } , \ldots , X _ { n } )$, $Y = ( Y _ { 1 } , \ldots , Y _ { n } )$ $Z = ( Z _ { 1 } , \ldots , Z _ { n } )$, $0 = ( 0 , \ldots , 0 )$. This group law on the sets $H ( B ) = \operatorname { nil } ( B ) ^ { n }$ is given by the formulas

\begin{equation} ( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } ) \end{equation}

where $z _ { i } = F _ { i } ( x _ { 1 } , \ldots , x _ { n } , y _ { 1 } , \ldots , y _ { n } )$; because $\pi$ and $y$ are nilpotent, all except a finite number of terms of the series are zero. Every formal group law gives group structures on $( B ) ^ { n }$ by means of

and converts the functor $D ^ { x }$ 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 $D$-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

Comments

A universal formal group law (for $12$-dimensional formal group laws) is an $12$-dimensional formal group law $F _ { u } ( X , Y ) \in L [ X , Y ]$, $X = ( X _ { 1 } , \ldots , X _ { n } )$, $Y = ( Y _ { 1 } , \ldots , Y _ { n } )$ such that for every $12$-dimensional formal group law $F ( X , Y )$ over a ring $4$ 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 $\frac { x } { 1 }$ to the coefficients of the $12$ 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 $I$ 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 $\alpha : F ( X , Y ) \rightarrow G ( X , Y )$, $\operatorname { dim } F = n$, $\operatorname { dim } G = m$, is an $m$-tuple of power series in $12$-variables $\alpha ( Z _ { 1 } , \ldots , Z _ { n } )$, $\alpha _ { i } ( 0 ) = 0$, such that $\alpha ( F ( X , Y ) ) = G ( \alpha ( X ) , \alpha ( Y ) )$. The homomorphism is an isomorphism if there exists an inverse homomorphism $3$ 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 $4$ be a ring of characteristic zero, i.e. the homomorphism of rings $Z \rightarrow A$ which sends $1 \in Z$ to the unit element in $4$ is injective. Then $A \rightarrow A \otimes z Q$ is injective. Over $A \otimes z Q$ all commutative formal group laws are strictly isomorphic and hence isomorphic to the additive formal group law

\begin{equation} G _ { \alpha } ( X , Y ) = ( X _ { 1 } + Y _ { 1 } , \ldots , X _ { n } + Y _ { n } ) \end{equation}

It follows that for every commutative formal group law $F ( X , Y )$ over $4$ there exists a unique $12$-tuple of power series $f ( X )$, $f _ { i } ( X ) = X _ { i } + \ldots$ with coefficients in $A \otimes z Q$ such that

\begin{equation} F ( X , Y ) = f ^ { - 1 } ( f ( X ) + f ( Y ) ) \end{equation}

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

\begin{equation} \operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i } \end{equation}

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 $A U ^ { * } ( n t )$. Cf. Cobordism for more details.

Let $F ( X , Y )$ be an $12$-dimensional group law over $4$. A curve over $4$ in $H ^ { \prime }$ is an $12$-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 $C ( F , A )$. The group $C ( F , A )$ admits a number of operators $V _ { n }$, $F _ { n }$, $[ a ]$, $x \in A$, defined as follows:

\begin{equation} V _ { n } \gamma ( T ) = \gamma ( T ^ { x } ) \end{equation}

\begin{equation} [ a ] \gamma ( T ) = \gamma ( a T ) \end{equation}

where $S _ { n }$ is a primitive $12$-th root of unity. There are a number of relations between these operators and they combine to define a (non-commutative) ring $( A )$, which generalizes the Dieudonné ring, cf. Witt vector for the latter. Cartier's second and third theorems on formal group laws say that the $( A )$ modules $C ( F , A )$ classify formal groups and they characterize which groups occur as $C ( F , A )$'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 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 $4$. Then $W ( A ) = C ( G _ { W } ; A )$. Cartier's first theorem for formal group laws says that the functor $F \mapsto C ( F ; A )$ is representable. More precisely, let $\hat { v }$ be the (infinite-dimensional) formal group law given by the addition formulas of the Witt vectors and let $\gamma _ { 0 } ( T )$ be the curve $( T , 0,0 , \ldots )$. Then for every formal group law $F ( X , Y )$ and curve $\gamma ( T ) \in C ( F ; A )$ there is unique homomorphism of formal group laws $\alpha _ { \gamma } : \hat { W } \rightarrow F$ 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 $4$ 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

\begin{equation} f _ { \pi } ( X ) = X + \pi ^ { - 1 } X ^ { q } + \pi ^ { - 2 } X ^ { q ^ { 2 } } + \end{equation}

Then $F _ { \pi } ( X , Y ) = f \pi ^ { 1 } ( f \pi ( X ) + f _ { \pi } ( Y ) )$ has its coefficients in $4$. These formal group laws are in a sense formal $D$-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 $4$. Indeed, let $F _ { \pi } ( \overline { m } )$ be the set $\overline { 112 }$ with the addition $\alpha + b = F _ { \pi } ( \alpha , b )$. Here $\overline { 112 }$ 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 $F ( \overline { m } )$; 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: $D$-divisible groups; cf. $D$-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 $H ^ { \prime }$ be defined over a field $k$ of characteristic $p > 0$. Then $[ p ] ( X )$ is necessarily of the form $X ^ { p ^ { k } } +$ (higher degree terms) or is equal to zero. The positive integer $h$ is called the height of $H ^ { \prime }$; if $[ p ] ( X ) = 0$, the height of $H ^ { \prime }$ 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 $1,2 , \ldots , \infty$ occur.

Let $F ( X , Y )$ be a one-dimensional formal group law over a ring $4$ in which every prime number except $D$ is invertible, e.g. $4$ is the ring of integers of a local field of residue characteristic $D$ or $4$ is a field of characteristic $D$. Assume for the moment that $4$ is of characteristic zero and let

\begin{equation} f ( X ) = X + \alpha _ { 2 } X ^ { 2 } + \alpha _ { 3 } X ^ { 3 } + \ldots \end{equation}

be the logarithm of $F ( X , Y )$. Then $F ( X , Y )$ is strictly isomorphic over $4$ to the formal group law $F _ { ( p ) } ( X , Y )$ whose logarithm is equal to

\begin{equation} f ( p ) ( X ) = X + a _ { p } X ^ { p } + a _ { p } 2 X ^ { p ^ { 2 } } + \end{equation}

The result extends to the case that $4$ is not of characteristic zero and to more-dimensional commutative formal group laws. $F _ { ( p ) } ( X , Y )$ is called the $D$-typification of $F ( X , Y )$.

References

[a1]	J.T. Tate, "$D$-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 $D$-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