In this paper we develop methods for classifying Baker, Richter, and 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, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two 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 $\mathbb {Z}/8\times \mathbb {Z}/2$. 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 $\mathrm {End}_{KO}(KU)$. 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.

在本文中，我们开发了在可交换环谱上对 Baker、Richter 和 Szymik 的 Azumaya 代数进行分类的方法，尤其是在环是非连通的大部分不可访问的情况下。我们提供了阻塞理论工具，用 Goerss-Hopkins 阻塞理论构造和分类这些代数及其自同构，并提供了下降理论工具，应用 Lurie 在$\infty$ -categories上的工作来证明环的有限伽罗瓦扩展在Rognes 的意义成为 Brauer 空间上的同伦不动点等价。对于偶周期环谱$E$，我们发现系数环是射影的“代数”Azumaya 代数受$\pi _0(E)$的 Brauer-Wall 群​​支配，恢复 Baker、Richter 和 Szymik 的结果。这使我们可以计算许多示例。例如，我们发现 Lubin-Tate 谱上的代数 Azumaya 代数有四个或两个 Morita 等价类，这取决于质数是奇数还是偶数，所有在复 K 理论谱$KU$上的代数 Azumaya 代数都是Morita 微不足道，并且本地化$KU[1/2]$上代数 Azumaya 代数的 Morita 类群是$\mathbb {Z}/8\times \mathbb {Z}/2$。使用我们的下降结果和阻塞理论谱序列，我们还研究了真实 K 理论谱$KO$上的Azumaya 代数，该谱变成了 Morita-trivial $KU$-代数。我们表明，它们恰好存在两个 Morita 等价类。非平凡的 Morita 等价类由具有与$\mathrm {End}_{KO}(KU)$相同的系数环的“奇异” $KO$ -代数实现。这需要仔细分析$KU$的 Picard 空间的同伦定点谱序列中发生的情况，之前由 Mathew 和 Stojanoska 研究过。