Studying the limit of a sequence of spectra knowing their BP-Homology

Question

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

Answer 1

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 $d_n$.