Brauer groups and Galois cohomology of commutative ring spectra

David Gepner, Tyler Lawson

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 ∞-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 π0(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 ℤ/8 × 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 EndKO(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.

Original languageEnglish (US)
Pages (from-to)1211-1264
Number of pages54
JournalCompositio Mathematica
Volume157
Issue number6
DOIs
StatePublished - Jun 2021

Bibliographical note

Publisher Copyright:
© 2021 Foundation Compositio Mathematica.

Keywords

  • Galois cohomology
  • Picard and Brauer groups
  • obstruction theory

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