QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments.

Let $X_n$ be the spectrum such that $BP_*(X_n) = BP_*/(v_0^{p^{i_0}},v_1^{p^{i_1}},\dots,v_{n-1}^{p^{i_{n-i}}})$. Recall that it's always possible to have a sequence $i_0, i_1, i_2 \dots$ such that a spectrum with the given BP-homology exists (this was my mistake before editing the question).

Let also $X = \bigvee_i X_i$.

Now consider the following fibration:

$ \bigvee_i X_i \to L_n \bigvee X_i \to \Sigma C_n \bigvee X_i $

which gives me the following inverse system of short exact sequences:

$$ 0 \to \bigoplus_{i \leq n} BP_*(X_i) \longrightarrow \ \bigoplus_{i \leq n} BP_*L_n(X_i) \ \longrightarrow \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to 0$$

$$ 0 \to \bigoplus_{i \leq n-1} BP_*(X_i) \to \bigoplus_{i \leq n-1} BP_*L_{n-1}(X_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{n-1}X_i) \to 0$$

with vertical maps $$ \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{i-1}X_i)$$

So here is my second question: How can i study the inverse limit of the right vertical tower $$lim_n \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i)?$$

Thank you

QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments.

Let $X_n$ be the spectrum such that $BP_*(X_n) = \Sigma^{d_n}BP_*/(v_0^{p^{i_0}},v_1^{p^{i_1}},\dots,v_{n-1}^{p^{i_{n-i}}})$. Recall that it's always possible to have a sequence $i_0, i_1, i_2 \dots$ such that a spectrum with the given BP-homology exists (this was my mistake before editing the question). $d_n$ is a natural number big enough which makes $X_i$ be a suspension spectrum.

Let also $X = \bigvee_i X_i$.

Now consider the following fibration:

$ \bigvee_i X_i \to L_n \bigvee X_i \to \Sigma C_n \bigvee X_i $

which gives me the following inverse system of short exact sequences:

$$ 0 \to \bigoplus_{i \leq n} BP_*(X_i) \longrightarrow \ \bigoplus_{i \leq n} BP_*L_n(X_i) \ \longrightarrow \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to 0$$

$$ 0 \to \bigoplus_{i \leq n-1} BP_*(X_i) \to \bigoplus_{i \leq n-1} BP_*L_{n-1}(X_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{n-1}X_i) \to 0$$

with vertical maps $$ \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{i-1}X_i)$$

So here is my second question: How can i study the inverse limit of the right vertical tower $$lim_n \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i)?$$

Thank you

Let $X_i$ be the spectrum such that $BP_*(X_i) = \Sigma^{i+1} BP_* / (v_0^2, v_1^2 , \dots, v_{i-1}^2)$ and $X = \bigvee_i X_i$.

First question: Can i always say that $X$ and $X_i$ are suspension spectra? Why?

Now let's consider the following fibration:

$ \bigvee_i X_i \to L_n \bigvee X_i \to \Sigma C_n \bigvee X_i $

which gives me the following inverse system of short exact sequences:

$$ 0 \to \bigoplus_{i \leq n} BP_*(X_i) \longrightarrow \ \bigoplus_{i \leq n} BP_*L_n(X_i) \ \longrightarrow \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to 0$$

$$ 0 \to \bigoplus_{i \leq n-1} BP_*(X_i) \to \bigoplus_{i \leq n-1} BP_*L_{n-1}(X_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{n-1}X_i) \to 0$$

with vertical maps $$ \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{i-1}X_i)$$

So here is my second question: How can i study the inverse limit of the right vertical tower $$lim_n \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i)?$$

Thank you

QUESTION EDITED: There was a mistake, the spectrum i had written before didn't even exist, so a big thanks to the people who made me notice that in the comments.

Let $X_n$ be the spectrum such that $BP_*(X_n) = BP_*/(v_0^{p^{i_0}},v_1^{p^{i_1}},\dots,v_{n-1}^{p^{i_{n-i}}})$. Recall that it's always possible to have a sequence $i_0, i_1, i_2 \dots$ such that a spectrum with the given BP-homology exists (this was my mistake before editing the question).

Let also $X = \bigvee_i X_i$.

Now consider the following fibration:

$ \bigvee_i X_i \to L_n \bigvee X_i \to \Sigma C_n \bigvee X_i $

which gives me the following inverse system of short exact sequences:

$$ 0 \to \bigoplus_{i \leq n} BP_*(X_i) \longrightarrow \ \bigoplus_{i \leq n} BP_*L_n(X_i) \ \longrightarrow \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to 0$$

$$ 0 \to \bigoplus_{i \leq n-1} BP_*(X_i) \to \bigoplus_{i \leq n-1} BP_*L_{n-1}(X_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{n-1}X_i) \to 0$$

with vertical maps $$ \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i) \to \bigoplus_{i \leq n-1} BP_{*-1}(\Sigma C_{i-1}X_i)$$

So here is my second question: How can i study the inverse limit of the right vertical tower $$lim_n \bigoplus_{i \leq n} BP_{*-1}(\Sigma C_iX_i)?$$

Thank you