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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

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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 $BP_*$.

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    $\begingroup$ @Tom That's not the case, e.g. when $n=0$, $X = \Bbb{RP}^2$, and $m=2$. The localizations $L_m$ and $C_m$ typically have some infinitely $v_d$-divisible torsion contributions for $n < d \leq m$ that one has to account for as well. $\endgroup$
    – Tyler Lawson
    Commented Dec 4, 2017 at 16:02

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