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

Abstract: We study the decategorification process that takes an Azumaya algebra to its $K(1)$-local $K$-theory. We prove various injectivity statements and relate precisely certain Brauer groups (more precisely, spectra) to certain $K$-theoretic strict Picard spectra or strict unit spectra. For example, we prove that for fields $F$ of characteristic $\neq p$, $\mathbf{Br}(F)[p^{\infty}] \simeq \mathbb{G}{\mathrm{pic}}(L{K(1)}K(F)\otimes \mathbb{S}{W(\overline{\mathbb F_p})})[p^\infty]$ where $\mathbb{G}{\mathrm{pic}}$ is Carmeli's strict Picard spectrum, and $\mathbf{Br}(F)$ denotes the Brauer space of the field $F$.