$K(1)$-local $K$-theory of Azumaya algebras
Maxime Ramzi
TL;DR
This work develops a detailed link between Brauer/strict Picard data and $K(1)$‑local K‑theory for Azumaya algebras, using strict unit/picard spectra, chromatic Fourier transform, and cyclotomic redshift to establish Künneth‑type results. A central achievement is a $K(1)$‑local Künneth formula for Azumaya algebras split by $p$‑group Galois covers, together with local‑to‑global identifications that relate $p$‑power torsion Brauer classes to strict Picard data after appropriate Witt vector renormalization. The paper also provides concrete computations in real KO settings (e.g., quaternions and Clifford algebras) and derives explicit identifications for the prime‑power torsion in $K$‑theory of number rings, demonstrating the practical impact on arithmetic and categorical K‑theory. Overall, the results illuminate how $K(1)$‑local K‑theory encodes Brauer and Picard information, offering new tools for understanding Morita theory, descent, and arithmetic phenomena in a chromatic context.
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 $\neq p$, $\mathbf{Br}(R)[p^\infty]\simeq \mathbb{G}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes\mathbb{S}_{W(\overline{\mathbb F}_p)})[p^\infty]$, 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.