TITLE: Morava $K$-theory of $K( \mathbb{Z}/p^2)$
QUESTION [10 upvotes]: 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.)

REPLY [11 votes]: $L_{K(1)}K({\Bbb Z}/p^n)$ 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.