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Let $E$ be a ring spectrum, and $X, Y$ spectra. What can we say about $E_*(X \wedge Y)$ from knowledge of $E_*(X), E_*(Y)$? Ideally I would hope that there would be some sort of Kunneth spectral sequence, for instance there is one in K-theory by a result of Atiyah. It would seem that the necessary condition is being able to embed a space in spaces whose $E_*$-homology is projective or something like that.

(Wikipedia indicates that I should look at Elmendorff-Kriz-Mandell-May, but I wonder if there is something which works for just plain ring spectra.)

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I'll add more to this later, but:

As far as the classical Kunneth formula, this is a very special thing to ask for. I think that essentially the only spectra that satisfy such a thing are like the Morava $K$-theories and the Eilenberg-Maclane spectra over fields (or PIDs...). (So for example, complex $K$-theory is special because it's determined by all the $K(1)$ theories.)

For a spectral sequence I'm not sure off the top of my head, but I'll get back to you later tonight when I have a moment!

EDIT: Actually EKMM do a very good job of describing the history of such results on page 32 of http://www.math.uchicago.edu/~may/PAPERS/Newfirst.pdf , as you suspected.

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Hmmm, ok. Is there some reason the Morava K-theories are so special in this way, though? –  Akhil Mathew Dec 31 '11 at 15:18
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Well, they are like the prime "fields" in the stable homotopy category; in particular, $K(n)_*E$ is always a free module over $K(n)_*$ for any spectrum $E$. Actually, now that I think about it, any (associative ring) spectrum $E$ that has the property that $E \wedge F$ splits up as a wedge up copies of shifts of $E$ will satisfy the Kunneth theorem. (This type of spectrum is usually called a "field".) –  Dylan Wilson Dec 31 '11 at 16:27
    
It's worth noting that these are not necessarily highly structured ring spectra we're talking about! I've made that mistake before... –  Dylan Wilson Dec 31 '11 at 16:29
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@Dylan: Sorry I misread what you were saying. Morava K-theories and homology with field coefficients are essentially the only homology theories satisfies $E_*(X \times Y) \simeq E_*(X) \otimes_{E_*(pt)} E_*(Y)$ –  Juan S Jan 1 '12 at 22:27
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Really you can just say "Morava K-theories $K(n)$" (for $n \in [0,\infty]$), which is slightly more satisfying. –  Aaron Mazel-Gee Apr 29 '12 at 22:42
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