In mathematics, Lazard's universal ring is a
ring introduced by
Michel Lazard in 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
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
defined as follows. We let
:
be
:
for indeterminates
, and we define the universal ring ''R'' to be the commutative ring generated by the elements
, 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 homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
s 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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
(over the integers) on generators of degree 1, 2, 3, ..., where
has degree
. proved that the coefficient ring of
complex cobordism is naturally isomorphic as a
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but ...
to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that
has degree
, because the coefficient ring of complex cobordism is evenly graded.
References
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*{{citation , mr=0253350, authorlink=Daniel Quillen, last= Quillen, first= Daniel , title=On the formal group laws of unoriented and complex cobordism theory, journal=
Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
, volume=75 , issue=6, year=1969 , pages=1293–1298 , doi=10.1090/S0002-9904-1969-12401-8 , doi-access=free
Algebraic topology
Algebraic groups
Algebraic number theory