Lazard's universal ring

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined.

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)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring.

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