Lazard's universal ring
There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let
- F(x, y)
- x + y + Σci,j xiyj
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 every 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 degree 1, 2, 3, ... (where ci,j has degree (i + j − 1)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that c i, j has degree 2(i+j-1), because the coefficient ring of complex cobordism is evenly graded.
- Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9
- Lazard, Michel (1955), "Sur les groupes de Lie formels à un paramètre", Bulletin de la Société Mathématique de France, 83: 251–274, ISSN 0037-9484, MR 0073925
- Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics, 443, Berlin, New York: Springer-Verlag, ISBN 978-3-540-07145-7, MR 0393050, doi:10.1007/BFb0070554
- Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bulletin of the American Mathematical Society, 75: 1293–1298, MR 0253350, doi:10.1090/S0002-9904-1969-12401-8