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Brauer groups and Galois cohomology of commutative ring spectra

Part of: Homotopy theory Operations and obstructions Division rings and semisimple Artin rings Homological methods (field theory)

Published online by Cambridge University Press:  04 June 2021

David Gepner  and
Tyler Lawson
David Gepner
Affiliation:
Department of Mathematics, Statistics, and Computer Science, The University of Illinois at Chicago, 851 S. Morgan Street, 322 Science and Engineering Offices (M/C 249), Chicago, IL60607, USAgepner@uic.edu
Tyler Lawson
Affiliation:
School of Mathematics, The University of Minnesota, 206 Church St SE, Minneapolis, MN55455, USAtlawson@math.umn.edu
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Abstract

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.

Keywords

Picard and Brauer groupsobstruction theoryGalois cohomology

MSC classification

Primary: 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.) 55S35: Obstruction theory
Secondary: 12G05: Galois cohomology 16K50: Brauer groups
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 6 , June 2021 , pp. 1211 - 1264
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Copyright
© The Author(s) 2021

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Footnotes

D.G. was partially supported by NSF grants DMS-1406529 and DMS-1714273 and T.L. was partially supported by NSF grant DMS-1206008.

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