Brauer groups and Galois cohomology of commutative ring spectra

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