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



Studying the limit of a sequence of spectra knowing their BP-Homology
There are no spectra with the indicated $BP$-homology. The $BP$-homology of a spectrum is always a $BP\_\*BP$-comodule, and $BP\_\*/(v\_0^i,v\_1^j)$ only admits a comodule structure if $\eta\_R(v\_1)^j=v\_1^j\pmod{v\_0^i}$. Here $\eta\_R(v\_1)$ can be calculated from the relation
$$ \sum^F\_{i,j}t\_i\eta\_R(v\_j)^{p^i}x^{p^{i+j}} = 
 \sum^F\_{i,j}v\_it\_j^{p^i}x^{p^{i+j}}
$$
as explained in Section 4.3 of Ravenel's "Complex Cobordism and Stable Homotopy Groups of Spheres", for example: we get 
$$ \eta\_R(v\_1) = v\_1 + (p-p^p) t\_1. $$
(This is with Araki generators; the numbers are slightly different with Hazewinkel generators, but the overall picture is the same. Of course $v\_0=p$ here.)


From this we find that $\eta\_R(v\_1)^2\neq 0\pmod{v\_0^2}$, so $BP\_\*/(v\_0^2,v\_1^2)$ is not a comodule. However, $\eta\_1(v\_1)^p=0\pmod{v\_0^2}$, so $BP\_\*/(v\_0^2,v\_1^p)$ is a comodule. But it is still quite delicate to decide whether there is a spectrum with $BP\_\*(X)=BP\_\*/(v\_0^2,v\_1^p)$, and it is not known how to settle such questions for ideals involving $v\_0,\dotsc,v\_n$ when $n$ is large. The best that you can do is to say that there exists a sequence $i\_0,i\_1,i\_2,\dotsc$ and spectra $X\_n$ for all $n$ such that
$$ BP\_\*(X\_n) = BP\_\*/(v\_0^{p^{i\_0}},\dotsc,v\_{n-1}^{p^{i\_{n-1}}}). $$
This follows from the work of Hopkins and Smith on existence of $v\_n$-self maps. As $X\_n$ is a finite spectrum, you can also choose $d\_n$ such that $\Sigma^{d\_n}X\_n$ is a suspension spectrum. I guess that the minimum possible $d\_n$ grows quite rapidly with $n$, much faster than $n$ itself. I don't think that there are any results in the literature that would give a useful upper bound on $d\_n$.


