Confusion about p-completion in Adams spectral sequence

Question

In section 3.1 of Complex Cobordism and Stable Homotopy Groups of Spheres, Ravenel computes the homotopy groups of $MU$ using the Adams spectral sequence. He comes to the conclusion that the $E_2$ page is $C \otimes P(a_0, a_1,\dots)$ where $C=P(u_1,\dots)$ where the order of $a_i$ is $2p^n -2$ and the order of $u_i$ is 2i and there is no $u_i$ if $i=p-1$.

Since the spectral sequence is concentrated in even degrees, this is also the $E_ \infty$ page. So this should be the associated graded for the homotopy groups of the p-completion of $MU$.

Now I believe these are polynomial rings over $F_p$ and tensor products over $F_p$. So how is it that the homotopy groups of $MU$ are free $\mathbb{Z}$-modules, but the associated graded of the p-completion contains no copies of the p-adic integers? I guess there is an extension $0 \rightarrow \mathbb{Z}_p \rightarrow \mathbb{Z}_p \rightarrow F_p \rightarrow 0$ since p-completion is exact. Is this the reason no copies of the p-adics show up?

Answer 1

In fact the extension $0 \rightarrow \mathbb{Z}_p \rightarrow \mathbb{Z}_p \rightarrow F_p \rightarrow 0$ is the the relevant extension. Unlike something like the Serre spectral sequence where the problem of going from the associated graded to the ring itself is something that can be done by repeated finite extension (since the diagonals with the filtration quotients are nonzero only finitely often), the Adams spectral sequence can have infinitely many filtration quotients contributing to the same homotopy group.

In the case of $MU$, we get that these diagonals have infinitely many copies of $F_p$ (this is due to the $a_0$ element) which corresponds to a filtration of $\mathbb{Z}_p$ in which each level of the filtration is isomorphic to $\mathbb{Z}_p$ and the quotients are $\mathbb{F}_p$.

We can be sure that this is the extension because it can be shown that this $a_0$ can be found on the $E_2$ page of the ASS for $S^0$, and the action of it on any permanent cycle corresponds to multiplying a element in $pi_*(X)$ by $p$.

So although we only have copies of $\mathbb{F}_p$ on the $E_\infty$ page, these actually stack up to build copies of the p-adic integers, signaling there is no p-torsion in the uncompleted homotopy groups.