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It is known in classical stable homotopy theory that there is an equivalence

$$ L_{K(1)}S\simeq KU^{h\mathbb{Z}_p^\times} $$ which gives an especially convenient way to compute the K(1)-local sphere. Motivically, we have analogues of $KU$ and $K(1)$. These are given, respectively, by the motivic spectrum representing algebraic $K$-theory $KGL$ and the algebraic Morava $K$-theory $K(1)$ as constructed by Borghesi. I am under the impression that the Adams operations also act $KGL$ via $E_\infty$-maps, and so it makes sense to write $KGL^{h\mathbb{Z}_p^\times}$. Is it known if there is analogous equivalence

$$ L_{K(1)}S^{0,0}\simeq KGL^{h\mathbb{Z}_p^\times}? $$

A more down to earth version of this question might be the following. Classically, we also have (in the 2-complete category) the Adams-Baird fibre sequence

$$ S_{K(1)}\to KO\to KO $$ where the map $KO\to KO$ is $\psi^3-1$. Motivically, the analog of $KO$ is the motivic spectrum $KQ$ representing Hermitian $K$-theory. Is it known whether or not the corresponding fibre sequence exists motivically?

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  • $\begingroup$ I don't know, but in the references to arxiv.org/pdf/1002.5007.pdf (which is 9 years old), Ormsby writes that there's a document in preparation about the K(1)-local motivic sphere. Presumably he knows something about your question. $\endgroup$
    – skd
    Mar 27, 2019 at 23:54
  • $\begingroup$ @skd good to know! I'll be sure to shoot him an email. $\endgroup$
    – CWcx
    Mar 28, 2019 at 0:15

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