$K(1)$-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 Künneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\ne p$, $\mathbf{Br}(R)[p^{\infty }]\simeq {\mathbb{G}}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes {\mathbb{S}}_{W({\overline{\mathbb{F}}}_p)})[p^{\infty }]$, 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.

43 pages, comments welcome! v2 : Changed introduction, added the example of p-inversions of p-henselian rings following a suggestion by Dustin Clausen and Akhil Mathew, added some material on Clifford algebras