Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated
($BP\_\*(X) = \Sigma^d BP\_\*/I$, where $I$ has the form $I=(v\_0^{i\_0}, \dots , v\_n^{i\_n})$ such that the homology is a $BP\_\*(BP)$ coalgebra).


Let $C\_nX$ the fiber of the map $ X \to L\_nX$ and let $\Sigma C\_n X$ be its cofiber.


What can be said about the natural map $$BP\_\*(\Sigma C\_n X) \to BP\_\*(\Sigma C\_{n-1} X) ? $$


My guess would be that it's always injective, but i'm not entirely sure about it.


Thanks



Map between homology of spectra
For any $m$, there is a Kunneth spectral sequence
$$
Tor\_{BP\_\*} (K(m)\_\*, BP\_\*(X)) \Rightarrow K(m)\_\* X.
$$
For your $X$, $BP\_\*(X)$ is acted on nilpotently by $v\_m$ for $0 \leq m \leq n$, and so this spectral sequence starts with zero.


Thus $K(m)\_\* X = 0$ for $0 \leq m \leq n$, and so for any $d \leq n$ we have
$$
0 = L\_{K(0) \vee K(1) \vee \dots \vee K(d)} X = L\_d X,
$$
by a result of Ravenel's.
Thus the fiber $C\_d(X)$ of the map $X \to L\_d(X)$ is equivalent to $X$, and $C\_{n} X \to C\_{n-1} X \to X$ are all equivalences. In particular, we get an *isomorphism* on $BP\_\*$.


