Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$, there is the unit map $$ M \to L_{K(1)}M. $$ If $M$ is the p-completion of the algebraic K theory of a smooth projective variety $M$ over $C$, by Quillen-Lichtenbaum conjecture, the unit map $$ K^{alg}(X)_{\hat{p}} \to L_{K(1)}K^{alg}(X) $$ is $n$-connected for some natural number $n$. Can we extend this story to general $M$? That is, the unit map $ M \to L_{K(1)}M $ is $n$-connected for some natural number $n$?
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4$\begingroup$ What are “hotopogy groups?” $\endgroup$– Jason StarrCommented Nov 13, 2022 at 16:08
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$\begingroup$ It was a typo. Thanks for letting me know. $\endgroup$– FredyCommented Nov 13, 2022 at 16:26
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2$\begingroup$ There is still a typo in the title, and at least two more in the body of the question. Please proof-read it. $\endgroup$– Neil StricklandCommented Nov 13, 2022 at 16:53
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