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Consider a Moore spectrum $S^n/p$. Has the $K$-theory of the extended powers of $S^n/p$ been computed?

For some context: equivalently, I'd like to have the free $E_\infty$-algebra over $KU$ on the $KU$-module $KU/p$. If we replace $S^n/p$ by $S^n$, then we do know the answer because of the theory of power operations over $KU$. Over $H \mathbb{F}_p$, these computations played an important role in the proof of Nishida's nilpotence theorem.

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    $\begingroup$ McClure computes some information about this in the H-infinity book, if I recall correctly. I think he gets the mod p K-theory of an arbitrary extended power of anything, and I remember him being particularly interested in Moore spectra at some point to deal with K-theory mod p^n. $\endgroup$
    – Dylan Wilson
    Commented Nov 7, 2016 at 16:48
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    $\begingroup$ When $n$ is even I seem to recall that the elements $\theta^{(j)}(px)$ form a regular sequence in the free $\theta$-algebra on $x$ (you can pick off a coefficient which is a unit times $(\theta^{(j-1)} x)^{p}$ for $j > 0$), and so I think the result is the (completed) quotient of $\Bbb Z_p[\theta^{(j)} x]$ by the ideal generated by these elements. I don't know how explicit this can get. $\endgroup$
    – Tyler Lawson
    Commented Nov 7, 2016 at 17:36
  • $\begingroup$ Thank you for these references, @DylanWilson, and @TylerLawson! These sound exactly like what I was looking for. $\endgroup$
    – Akhil Mathew
    Commented Nov 8, 2016 at 20:37

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