Collapse of Hirzebruch Spectral sequence

Question

This question is actually about reading Adams' Stable Homotopy and Generalised Cohomology; in Part II chapter 2, there are two numbered lemmata (Lemma 2.5 contravariant, 2.14 covariant) to the effect that

The Atiyah-Hirzebruch spectral sequences $H_p(\mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty)$ and $H_p(\mathbb{CP}^\infty\times \mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty\times \mathbb{CP}^\infty)$ collapse at page 2.

Now, read in context, Adams has already mentioned that there are three spectra he's interested in: $H,KU,MU$ (where the AH-SS indeed collapses for sparsity reasons); but there is no mention of this in the statements of either lemma, nor in the (very terse) arguments given.

Question Can someone confirm whether $E\in \{ H, KU , MU \}$ is in fact what Adams meant?

Answer 1

I couldn't find the exact reference, but I guess $H$, $KU$ and $MU$ are probably in particular what is meant.

In general, the Atiyah–Hirzebruch spectral sequence collapses for complex oriented multiplicative cohomology theories (cf. e.g. Proposition 7 in Lecture 4 of Lurie's notes on chromatic homotopy theory).

Edit: Now that we have a more precise reference, I found that that in the beginning of that chapter it reads:

We will study [ring] spectra $Ε$ which are provided with "orientations", in the following sense (which owes much to a seminar by A. Dold). There is given an element $x \in \tilde{E}^*(\mathbb{C}P^\infty)$ such that $\tilde{E}^*(\mathbb{C}P^1)$ is a free module over $\pi_*(E)$ on the generator $i^*x$, where $i \colon \mathbb{C}P^1 \to \mathbb{C}P^\infty$ is the inclusion map.

So it looks like what Adams actually means is also a general complex oriented cohomology theory (resp. complex oriented spectrum).