Where can I find a proof of the Hodge-Tate decomposition for Lubin-Tate formal groups?
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I recommend reading Tate's original paper, which proves the Hodge--Tate decomposition for all $p$-divisible groups. If you are nervous about $p$-divisible groups, rather than formal groups, it would not be difficult to restrict to just this case while reading the paper. (And the paper includes an entire section devoted to relating formal groups to the $p$-divisible groups picture, which is valuable in its own right.)