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A universal formal group and complex cobordism

1975; American Mathematical Society; Volume: 81; Issue: 5 Linguagem: Inglês

10.1090/s0002-9904-1975-13891-2

1088-9485

Michiel Hazewinkel,

Homotopy and Cohomology in Algebraic Topology

The purpose of this note is to 'announce' some of the results of [5], [6], [7] pertaining to formal groups and complex cobordism.These should have been written up a number of years ago.The phrase "formal group" is used as an abbreviation for commutative one-dimensional formal group (law).1. Introduction.Below we give an explicit recursion formula for the logarithm of a universal commutative formal group and a p-typically universal commutative formal group.These give us a universal formal group F v defined over. .] and a/^-typically universal formal group F T over Z[T X , T 2 , . . .]. Possibly the best way to look at these formal groups is as follows.To fix ideas let p be a fixed prime number and let A be a commutative ring with unit such that every prime number =£ p is invertible in A. Let F T be the one-dimensional p-typically universal formal group and G a one-dimensional formal group over A. Cartier [4] associates to G a module of curves C(G) over a certain ring Cart p (4).The ring Cart p (/1) has as its elements expressions 2 V 1 [a t j] f 7 , a t j £ A, which are added and multiplied according to certain rules, cf.[4] and [9] ; V stands for the 'Verschiebung' associated to the prime number p and f stands for the 'Frobenius' associated to the prime number p.The left modules C over Cart p (/4) which arise as modules of curves of some one-dimensional commutative formal group are of the formNow let F t be the formal group over A obtained by substituting t t for T v Then OF t ) = C. 2. The formulae.Choose a prime number p and let (2.1) l n (T) = Z T h Tf? • • • T( 1+ '" +is ~lIpwhere the sum is over all sequences (i 1 , i 2 , . . ., Q, i-E N = {1, 2, 3, . . .} such that ij + • • • + jy = n.