| Formal Group Law |
Article Index for Formal |
Website Links For Formal |
Information AboutFormal Group Law |
F(x1,x2, ..., xn,y1,y2, ..., yn) where ''n'' is the dimension of ''G'', taking values in R''n''. Suppose we take the Power Series expansion of ''F'' (formal Taylor Series ) and look at the identities it should satisfy on the basis of the group axioms - for example the Associative Law (for the inverse operation take also an expansion of the inverse mapping ''I''). Then we can turn these power series identities into a fresh definition, of a '''formal group law''' (often just '''formal group''') in ''n'' variables. The nature of a 'formal group' is not therefore as a group in the set-theoretic sense: it is more like the bare definition F(x,y) = x + y + xy that comes from thinking about multiplication near 1. It is not either a Group Object in the sense of category theory (as a Lie Group is). It does provide an actual group when the variable ''x'' is given values in a ring such as :R {Link without Title} /(''tk''), for ''k'' = 1, 2, ... for which convergence is guaranteed since ''t'' has been forced to be Nilpotent . There is an abstract sense in which the formal group here is an Inverse Limit of Group Scheme s; or a Functor from Artinian R-algebras to groups. There was early work on formal groups of Elliptic Curve s by Lutz; the idea later became important in the classification of Abelian Varieties in non-zero characteristic. The theory was also much used in Algebraic Topology in the 1960s, after it was seen that it played a role in Homology Theory . To be precise, Cobordism Theory needed the ''universal'' one-dimensional formal group over Z, the existence of which had been proved by Lazard. |
|
|