$K(1)$-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 $F$.

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