Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th