Формальды топтар заңы: математикалық анықтамалар мен қолданыстары

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 introduced by 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.

When F is one dimensional, one can write its logarithm in terms of the invariant differential ω(t). Let where is the free module of rank 1 on a symbol dt. Then ω is translation invariant in the sense that where if we write , then one has by definitionIf one then considers the expansion , the formuladefines the logarithm of F.

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 more involved. 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) = ΣD(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 xiyj 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.

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 εg = 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).

A formal group is a group object in the category of formal schemes. If is a functor from Artin algebras to groups which is left exact, then it is representable (G is the functor of points of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits). If is a group scheme then , the formal completion of G at the identity, has the structure of a formal group. The formal completion of a smooth group scheme is isomorphic to Some people call a formal group scheme smooth if the converse holds; others reserve the term "formal group" for objects locally of this form. Formal smoothness asserts the existence of lifts of deformations and can apply to formal schemes that are larger than points. A smooth formal group scheme is a special case of a formal group scheme. Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections. The (non strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group. Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object. The moduli space of formal group laws is a disjoint union of infinite dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite dimensional groupoid of coordinate changes. Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p divisible groups, Dieudonné theory, and Galois representations. For example, the Serre Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations. A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added, such as being pointed or connected). This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring. Some authors use the term formal group to mean formal group law.

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

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 , in a successful effort to isolate the local field part of the classical theory of complex multiplication of elliptic functions. It is also a major ingredient in some approaches to local class field theory and an essential component in the construction of Morava E theory in chromatic homotopy theory.