Brown–Peterson cohomology

Template:Short description Template:No footnotes 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 $\text{BP}$.

Complex cobordism and Quillen's idempotent

Brown–Peterson cohomology $\text{BP}$ 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 $\text{BP}$.

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 ${ϵ}_{\ast }([C{P}^n])$ is [CP^$n$] if $n+1$ is a power of $p$, and $0$ otherwise. The spectrum $\text{BP}$ is the image of this idempotent ε.

Structure of $\text{BP}$

The coefficient ring ${\pi }_{\ast }(\text{BP})$ is a polynomial algebra over ${\mathbb{Z}}_{(p)}$ on generators $v_n$ in degrees $2(p^n-1)$ for $n\ge 1$.

${\text{BP}}_{\ast }(\text{BP})$ is isomorphic to the polynomial ring ${\pi }_{\ast }(\text{BP})[t_1,t_2,\dots ]$ over ${\pi }_{\ast }(\text{BP})$ 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 $({\pi }_{\ast }(\text{BP}),{\text{BP}}_{\ast }(\text{BP}))$ is the initial term of the Adams–Novikov spectral sequence for calculating $p$-local homotopy groups of spheres.

$\text{BP}$ 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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