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?



Collapse of Hirzebruch Spectral sequence
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](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf) 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).


