Formal group

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In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were first defined in 1946 by S. Bochner. The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.

Contents

[edit] Definitions

A one-dimensional formal group law over a commutative ring R is a power series F(x,y) with coefficients in R, such that

  1. F(x,y) = x + y + terms of higher degree
  2. F(x, F(y,z)) = F(F(x,y), z) (associativity)

The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.

More generally, an n dimensional formal group law is a collection of n power series Fi(x1, x2, ..., xn, y1, y2, ..., yn) in 2n variables, such that

  1. F(x,y) = x + y + terms of higher degree
  2. F(x, F(y,z)) = F(F(x,y), z)

where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on.

The formal group law is called commutative if F(x,y) = F(y,x). One dimensional formal group laws are usually commutative; in fact they are all commutative unless the ring R contains a non-zero nilpotent torsion element.

There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0.

A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that

G(f(x), f(y)) = f(F(x, y)).

A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)= x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a "change of coordinates".

[edit] Examples

  • The additive formal group law is given by
F(x,y) = x + y
  • The multiplicative formal group law is given by
F(x,y) = x + y + xy

This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b) = ab. If we "change coordinates" to make 0 the identity by putting a=1+x, b=1+y, and G=1+F, then we find that F(x,y) = x + y + xy. Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x)-1. Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic.

  • More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of dimension n, by taking coordinates at the identity and writing down the formal power series expansion of the product map. The additive and multiplicative formal group laws are obtained in this way from the additive and multiplicative algebraic groups. Another important special case of this is the formal group (law) of an elliptic curve (or abelian variety).
  • F(x,y) = (x + y)/(1+ xy) is a formal group law coming from the addition formula for the hyperbolic tangent funtion: tanh(x+y) = F(tanh(x), tanh(y)), and is also the formula for addition of velocities in special relativity (with the speed of light equal to 1).
  • F(x,y) = (x\sqrt{1-y^4} +y\sqrt{1-x^4})/(1+x^2y^2) is a formal group law over Z[1/2] found by Euler, in the form of the addition formula for an elliptic integral:
\int_0^x{dt\over \sqrt{1-t^4}} + \int_0^y{dt\over \sqrt{1-t^4}} = \int_0^{F(x,y)}{dt\over \sqrt{1-t^4}}

[edit] Lie algebras

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F2 of the formal group law.

[x,y] = F2(x,y) − F2(y,x)

The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group:

Lie groups → Formal group laws → Lie algebras

Over fields of characteristic 0, formal group laws are essentially the same as finite dimensional Lie algebras: more precisely, the functor from finite dimensional formal group laws to finite dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p>0.

[edit] The logarithm of a commutative formal group law

If F is a commutative n-dimensional formal group law over a commutative ring R containing the rational numbers Q, then it is strictly isomorphic to the additive formal group law. In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that

f(F(x,y)) = f(x) + f(y)

This is still true if R does not contain the rationals, except that f is now an isomorphism of formal groups over R⊗Q (and not over R), and has coefficients in R⊗Q rather than in R.

Formal group laws over a ring R are often constructed by first writing down their logarithm as a power series with coefficients in R⊗Q, and then proving that the coefficients of the corresponding formal group over R⊗Q actually lie in R.

Examples:

  • The logarithm of F(x,y) = x+y is f(x) = x.
  • The logarithm of F(x,y) = x+y+xy is f(x) = log(1+x).

[edit] The formal group ring of a formal group law

The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ring of a group and to the universal enveloping algebra of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups.

For simplicity we describe the 1-dimensional case; the higher dimensional case is similar except that notation becomes messier.

Suppose that F is a (1 dimensional) formal group law over R. Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows.

  • As an R-module, H is free with a basis 1=D(0), D(1), D(2),...
  • The coproduct Δ is given by ΔD(n) = ∑D(i)⊗D(n−i) (so the dual of this coalgebra is just the ring of formal power series).
  • The counit η is given by the coefficient of D(0).
  • The identity is 1=D(0).
  • The antipode S takes D(n) to (-1)nD(n).
  • The coefficient of D(1) in the product D(i)D(j) is the coefficient of x(i)y(j) in F(x,y).

Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.

[edit] Formal group laws as functors

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is Nn where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms. This makes F into a functor from commutative R-algebras S to groups.

We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups. For example, this allows us to define F(Zp) with values in the p-adic numbers.

The group valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg=g⊗g and ηg=1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form

D(0)+D(1)x + D(2)x2+...

for nilpotent elements x. In particular we can identify the group-like elements of H⊗S with the nilpotent elements of S, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S).

[edit] The height of a formal group law

Suppose that f is a homomorphism between one dimensional formal group laws over a field of characteristic p>0. Then f is either zero, or the first nonzero term in its power series exansion is ax^{p^h} for some non-negative integer h, called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞.

The height of a one dimensional formal group law over a field of characteristic p>0 is defined to be the height of its pth power map.

Two one dimensional formal group laws over an algebraically closed field of characteristic p>0 are isomorphic if and only if they have the same height, and the height can be any positive integer or ∞.

Examples:

  • The additive formal group law F(x,y) = x + y has height ∞, as its pth power map is 0.
  • The multiplicative formal group law F(x,y) = x + y +xy has height 1, as its pth power map is (1+x)p - 1 = xp.

[edit] The Lazard ring

There is a universal commutative one dimensional formal group law over a universal commutative ring defined as follows. We let F(x,y) be x +y +Σci,j xiyj for indeterminates ci,j, and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:

For any commutative ring S, one dimensional formal group laws over S correspond to ring homomorphisms from R to S.

The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, … (where ci,j has degree 2(i+j−1)). Daniel Quillen proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring, explaining the unusual grading.

[edit] Generalizations

There is no standard definition of the term formal group. The term formal group sometimes just means formal group law, and sometimes means one of several generalizations. Some possible generalizations include:

Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutative rings or fields.

[edit] Lubin-Tate formal group laws

We let Zp be the ring of p-adic integers. The Lubin-Tate formal group law is the unique (1 dimensional) formal group law F such that e(x)= px + xp is an endomorphism of F, in other words

e(F(x,y)) = F(e(x), e(y))

More generally we can allow e to be any power series such that e(x)= px + higher degree terms and e(x) =xp mod p. All the group laws for different choices of e satisfying these conditions are strictly isomorphic.

For each element a in Zp there is a unique endomorphism f of the Lubin-Tate formal group law such that f(x) = ax + higher degree terms. This gives an action of the ring Zp on the Lubin-Tate formal group law.

There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field.

This construction was introduced by Jonathan Lubin and John Tate, in a successful effort to isolate the local field part of the classical theory of complex multiplication of elliptic functions.

[edit] See also

[edit] References

  • S. Bochner, Formal Lie groups, Ann of Math. 47 (1946) 192-201.
  • Formal group in the Encyclopaedia of Mathematics
  • Formal Groups and Applications (Pure and Applied Math 78) Michiel Hazewinkel Publisher: Academic Pr (June 1978) ISBN 0-12-335150-2
  • Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics) by J. F. Adams University Of Chicago Press; Reissue edition (February 27, 1995) ISBN 0-226-00524-0
  • J. Lubin, J. Tate, Formal complex multiplication in local fields Ann. of Math. , 81 (1965) pp. 380–387
  • N. Strickland, Formal groups
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