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Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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Morava $K$-theory of $K( \mathbb{Z}/p^2)$

$L_{K(1)}K({\Bbb Z}/p^n)$ 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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