Brown–Peterson cohomology

Remarks

The coefficient ring is $\pi_* BP \cong \Z_{(p)}[v_1, v_2, \ldots]$ with $\abs{v_i} = 2(p^i - 1)$. With Hazewinkel's generators, the logarithm coefficients satisfy $p \ell_n = \sum_{i=0}^{n-1} \ell_i v_{n-i}^{p^i}$. For example, $p \ell_1 = v_1$, so $\ell_1 = v_1 / p$. The formal group law of $BP$ is the universal $p$-typical formal group law over a $\Z_{(p)}$-algebra.

Example: $MU_{(p)}$ is equivalent to a wedge $\bigvee_j \Sigma^{2 d_j} BP$, so $BP$ is a retract of $MU_{(p)}$, as an image of an idempotent should be.

Non-example: $MU_{(p)}$ itself is not $BP$, because $\pi_2 MU_{(p)} \cong \Z_{(p)}$ is nonzero for every $p$, while $\pi_2 BP = 0$ for $p$ odd since the first generator $v_1$ has degree $2(p - 1) \ge 4$.

Brown and Peterson first built $BP$ in 1966 by a different construction; Quillen's idempotent gives the description above.