Brown–Peterson Cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by
, depending on a choice of prime ''p''. It is described in detail by .
Its representing spectrum 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 of suspensions of BP.

For each prime ''p'', Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ_(''p'') to itself, with the property that ε( P^''n'' is 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 $\pi_*(\text)$ is a polynomial algebra over $\Z_$ on generators $v_n$ in degrees $2(p^n-1)$ for $n\ge 1$.

$\text_*(\text)$ is isomorphic to the polynomial ring \pi_*(\text) _1, t_2, \ldots/math> over $\pi_*(\text)$ with generators $t_i$ in $\text_(\text)$ of degrees $2 (p^i-1)$.

The cohomology of the Hopf algebroid $(\pi_*(\text), \text_*(\text))$ is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres.

BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.

See also

* List of cohomology theories#Brown–Peterson cohomology

References

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Cohomology theories