Brauer groups and Galois cohomology of commutative ring spectra

TL;DR

This paper develops methods to classify Azumaya algebras over commutative ring spectra, especially nonconnective cases, using obstruction and descent theories, and applies these to compute examples over various spectra.

Contribution

It introduces new obstruction-theoretic and descent-theoretic tools for classifying Azumaya algebras over ring spectra, extending previous results to nonconnective cases and providing explicit computations.

Findings

Classified Azumaya algebras over Lubin-Tate spectra, finding 2 or 4 Morita classes.

Proved all algebraic Azumaya algebras over $KU$ are Morita trivial.

Identified two Morita classes of Azumaya algebras over $KO$ with one exotic class.

Abstract

In this paper we develop methods for classifying Baker-Richter-Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss-Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$, we find that the "algebraic" Azumaya algebras whose coefficient ring is projective are governed by the Brauer-Wall group of $\pi_0(E)$, recovering a result of Baker-Richter-Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin-Tate spectra have either 4 or 2 Morita equivalence classes depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\Bbb Z/8 \times \Bbb Z/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an "exotic" $KO$-algebra with the same coefficient ring as $End_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$, previously studied by Mathew and Stojanoska.