$K(1)$-local $K$-theory of Azumaya algebras

We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a Künneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\mathrm{\ne }p$, $\mathbf{B}\mathbf{r}(R)[p^{\mathrm{\infty }}]\simeq {\mathbb{G}}_{\mathrm{p}\mathrm{i}\mathrm{c}}(L_{K(1)}K(R)\otimes {\mathbb{S}}_{W({\bar{\mathbb{F}}}_p)})[p^{\mathrm{\infty }}]$, where ${\mathbb{G}}_{\mathrm{p}\mathrm{i}\mathrm{c}}$ is Carmeli’s strict Picard spectrum. We prove the same result for $R[\frac{1}{p}]$, when $R$ is $p$-henselian.