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!



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


