Descent for $K(1)$-local spectra

Question

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.

Answer 1

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.