For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by
$$L\_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i\_K ] {}}} K$$
where $g$ is a topological generator of $\mathbb{Z}\_p^{\times}$ and $\Psi^g$ denotes the Adams operation. Now we can apply the functor $Mod(-): \textrm{Spectra} \to \textrm{Symmetric monoidal infinity categories} $ and then apply the functor $L\_{K(1)}$. This gives us a diagram of the form
$$L\_{K(1)}Sp \rightarrow Mod\_{K(1)}(K){{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i\_K ] {}}} Mod\_{K(1)}(K)$$ where $L\_{K(1)}Sp$ are the $K(1)$-local spectra and $Mod\_{K(1)}(K)$ are the $K(1)$-local $K$ modules.
Now is this an equalizer diagram?


I have been told that this is most likely true, so I was just wondering if there was a reference for this statement. Thanks in advance.



Descent for $K(1)$-local spectra
It's not quite true: need to require a $p$-adic continuity condition for the $\Psi^g$-semilinear automorphism of the $K(1)$-local $K$-module. You can see <https://arxiv.org/pdf/2001.11622.pdf> Proposition 3.10 for a slight variant which also works at the prime 2.


