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

Abstract

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 K\"unneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\neq p$, $\mathbf{Br}(R)[p^\infty]\simeq \mathbb{G}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes\mathbb{S}_{W(\overline{\mathbb F}_p)})[p^\infty]$, where $\mathbb{G}_{\mathrm{pic}}$ is Carmeli's strict Picard spectrum. We prove the same result for $R[\frac{1}{p}]$, when $R$ is $p$-henselian.