Explicit $BP_*BP$-comodule structure on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$

Question

So as it says in the title, how can one explicitly calculate the comodule structures on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$ for a prime $p$?

For example, $\mathbb{C}P^2$ sits in a cofiber sequence of spectra

$$\Sigma^2\mathbb{S}\to \mathbb{C}P^2\to \Sigma^4\mathbb{S} $$

giving rise to a SES of $BP_*BP$-comodules

$$0\to\Sigma^2BP_*\to BP_*\mathbb{C}P^2\to\Sigma^4 BP_*\to 0 $$

which defines an element in $Ext_{BP_*BP}^{1,2}(BP_*,BP_*)\cong \mathbb{Z}/2$ (this is for $p=2$). The comodule structure on $BP_*(\Sigma^2\mathbb{S}\vee\Sigma^4\mathbb{S})$ is trivial, so the comodule structure on $BP_*\mathbb{C}P^2$ should be non-trivial.

I tried calculating explicitly what the comodule map should be using properties of $BP_*BP$ and the counital condition for comodule maps to get that $\Psi:BP_*\mathbb{C}P^2\to BP_*BP\otimes_{BP_*}BP_*\mathbb{C}P^2$ is given on $BP_*$-module generators by

$$\Psi(g_1) = 1\otimes g_1, \Psi(g_2)= 1\otimes g_2 + t_1\otimes g_1. $$

(Here, $BP_*\mathbb{C}P^2\cong\Sigma^2BP_*\{g_1\}\oplus \Sigma^4BP_*\{g_2\}$ as $BP_*$-modules and $BP_*BP\cong BP_*[t_1, t_2,\ldots]$ with $\vert t_i\vert=2p^i-2$). I could very well be making a mistake though. I am still gaining familiarity with these $BP_*BP$ calculations.

As $n$ gets bigger, doing it explicitly like this will get a lot more time-consuming it seems. Are there other ways to realize the comodule structure on $BP_*\mathbb{C}P^n$ and, more generally, on $BP_*\mathbb{C}P^{\infty}$?

Has this been figured out by anyone else? If so, a reference if it exists would be great!

Thanks!

Answer 1

A concise formula $$ \mu(\beta) = \beta(c(t^F)) $$ for the $BP_* BP$-coaction $\mu$ on $BP_* CP^\infty$ is given in the "Note added in proof" on page 279 of

Ravenel, Douglas C.; Wilson, W. Stephen
The Hopf ring for complex cobordism.
J. Pure Appl. Algebra 9 (1976/77), no. 3, 241–280.

A more explicit formula is given in Theorem 3.13 of

Wilson, W. Stephen
Brown-Peterson homology: an introduction and sampler.
CBMS Regional Conference Series in Mathematics, 48. (1982)

where the result is deduced from the $MU$ case, given in Theorem 1.48(e).