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

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.

💡 Research Summary

The paper investigates the relationship between K(1)-local algebraic K‑theory and Azumaya algebras, establishing a precise connection between the strict Picard spectrum of the K(1)-local K‑theory of a scheme and its Brauer group. The author begins by recalling that for a commutative ring spectrum R, the spectrum of strict units Gₘ(R)=Map_{Spcn}(ℤ,gl₁(R)) and its delooping, the strict Picard spectrum G_{pic}=Map_{Spcn}(ℤ,Pic), have become central objects in recent work on “flat algebraic geometry.” The paper’s central theme is that after K(1)-localization, many of the usual failures of Künneth formulas for K‑theory of Azumaya algebras disappear, at least for p‑primary torsion classes.

The main results are organized as follows:

Theorem A (Theorem 3.19, Corollaries 3.9, 3.10).
Let X be a qcqs scheme with p∈𝔾ₘ(X). The canonical map
 Pic(X)