Lazard's universal ring
Encyclopedia
In mathematics, Lazard's universal ring is a ring
introduced by 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
be
for indeterminates
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:
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)). proved that the coefficient ring of complex cobordism
is naturally isomorphic as a graded ring to Lazard's universal ring.
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
introduced by 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
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
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)). proved that the coefficient ring of complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
is naturally isomorphic as a graded ring to Lazard's universal ring.