# A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$

I am reading the paper Quadratic functions in geometry, topology and M theory by M.J.Hopkins and I.M.Singer, and in section 4.8 they say :

'Recall that for any compact $S$, and for any cohomology theory $E$ there is a canonical isomorphism $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$.When $E$ is $K$-theory, this isomorphism is given by the Chern character. The isomorphism arises from a universal cohomology class $i_{E}\in H^{0}(E;\pi_{\ast}E\otimes\mathbb{R}) = \varprojlim H^{n}(E_{n};\pi_{\ast+n}E\otimes\mathbb{R})$ and associates to a map $f:S\to E_{n}$ representing an element of $E^{n}(S)$ the cohomology class $f^{\ast}i_{n}$, where $i_{n}$ is the projection of $i_{E}$ to $H^{n}(E_{n};\pi_{\ast+n}E)$.

I do not understand most of the above paragraph. My specific questions are :

1.) Where can I find a proof of the isomorphism mentioned ? I am familiar with the basic definitions of spectra and generalized cohomology but can not quite understand the isomorphism. Is this some kind of Brown representability ?

2.) I do not understand the system of abelian groups over which the limit is taken.

Of course, it is unreasonable to ask for detailed explanations for my doubts. I would really appreciate some hints, and pointers to good texts where I can find a discussion of the above issues and educate myself. Thanks so much !

• I would check Adams's "Stable Homotopy and Generalised Homology". You might also like to check Rudyak's "On Thom Spectra, Orientability, and Cobordism". – Tyrone Jun 8 '18 at 9:25