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The $p$-adic completion of $K( \mathbb{F}_p)$ is known (by Quillen's calculation) to be $H \mathbb{Z}_p$; in particular, $K(\mathbb{F}_p)$ is acyclic with respect to all Morava $K$-theories $K(n), 0 < n < \infty$ at the implicit prime $p$.

Does this hold for $\mathbb{Z}/p^2$? That is, is $L_{K(1)} K( \mathbb{Z}/p^2)$ (with the implicit prime for $K(1)$ equal to $p$) nontrivial? (A theorem of Mitchell implies that all the higher $K(i)$-localizations are trivial.)

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    $\begingroup$ If I understand correctly, the relative $K$-theory (i.e. the fibre of the map $K(\mathbb{Z}/p^k)\to K(\mathbb{Z}/p)$) is the same as the relative $TC$, so you want to prove that the relative $TC$ is $K(1)$-acyclic. That sounds more tractable, but I do not remember enough about $TC$ to carry it out. $\endgroup$ Sep 16, 2015 at 17:51

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$L_{K(1)}K({\Bbb Z}/p^n)$ is always trivial. This is Proposition 2.14 in the recent preprint by Bhatt, Clausen and yourself. Alternatively, this also appears in recent work by Land, Meier and Tamme.

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