In mathematics, Brown–Peterson cohomology is a
generalized cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
introduced by
, depending on a choice of prime ''p''. It is described in detail by .
Its representing
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
is denoted by BP.
Complex cobordism and Quillen's idempotent
Brown–Peterson cohomology BP is a summand of MU
(''p''), which is
complex cobordism MU
localized at a prime ''p''. In fact MU
''(p)'' is a
wedge product
A wedge is a triangular shaped tool, a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a fo ...
of
suspensions of BP.
For each prime ''p'',
Daniel Quillen showed there is a unique
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
map of
ring spectra ε from MUQ
(''p'') to itself, with the property that ε(
''n''">P''n'' is
''n''">P''n''if ''n''+1 is a power of ''p'', and 0 otherwise. The spectrum BP is the image of this idempotent ε.
Structure of BP
The coefficient ring
is a polynomial algebra over
on generators
in degrees
for
.
is isomorphic to the polynomial ring