Brown–Peterson cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Template:Harvs, depending on a choice of prime p. It is described in detail by Template:Harvs. 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([CPⁿ]) is [CPⁿ] 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 ${\pi }_{*}(\text{BP})$ is a polynomial algebra over ${\mathbb{Z}}_{(p)}$ on generators $v_n$ in degrees $2(p^n-1)$ for $n\ge 1$.

${\text{BP}}_{*}(\text{BP})$ is isomorphic to the polynomial ring ${\pi }_{*}(\text{BP})[t_1,t_2,\dots ]$ over ${\pi }_{*}(\text{BP})$ with generators $t_i$ in ${\text{BP}}_{2(p^i-1)}(\text{BP})$ of degrees $2(p^i-1)$.

The cohomology of the Hopf algebroid $({\pi }_{*}(\text{BP}),{\text{BP}}_{*}(\text{BP}))$ 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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