TITLE: Map between homology of spectra
QUESTION [5 upvotes]: 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

REPLY [9 votes]: 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 $BP_*$.