Brown–Peterson cohomology

Summary

Brown–Peterson cohomology is a generalized cohomology theory attached to a prime $p$, represented by a ring spectrum $BP$. After complex cobordism $MU$ is localized at $p$, it splits as a wedge of suspensions of copies of $BP$. So $BP$ keeps all the $p$-local information of $MU$ in a much smaller object, with coefficient ring $\pi_* BP = \Z_{(p)}[v_1, v_2, \ldots]$ and $\abs{v_i} = 2(p^i - 1)$. It is the basic tool of the Adams–Novikov spectral sequence and of chromatic homotopy theory.