Could you be a bit more specific? I know the case of $R$-modules but I am unable to generalize the proof to the case of ringed spaces. One problem is that for sheaves the distinction between internal and external $Hom$ and $Ext$ has to be made.

Thank you for the answer. Is it possible to write down explicitely where an element of $H^n(H,CoInd_H^G(A))$ gets mapped to?

If one chooses such a splitting, how would you make this choice into an inverse for the Shapiro Lemma?

@tj_: Could you explain what the inverse is?

Ups my bad :) So we can get $H^*(X;\mathbb{Q})$ but only for finite CW-Complexes? In particular I don't see any of the theory of chern classes here? This is confusing to me, because I would start with the Chern classes to make $H^*(X;\mathbb{Q})$ into a $MU^*$-module in the first place.

I am aware that they are coming from spectra, but I am not sure how to translate one into the other. Landweber writes in the original paper that it gives a homology theory on all CW-spectra and Rudyak does the same. So there are versions with non-finite complexes. Also surely one would want to be able to talk about $\mathbb{C} P^\infty$ which is an infinite complex

Yes I meant what @Phil said. Treat the system as a system in the category of abelian groups, i.e. think of it as the image of the functor. Then every object maps to zero IN the direct system. Is the last line still wrong?

@pro Yes, see the answer I just posted. Please let me know if there is a mistake.