Commutative ring-spectra of characteristic 2

1986 ◽  
Vol 61 (1) ◽  
pp. 33-45 ◽  
Author(s):  
Urs Würgler
Keyword(s):  
Commutative Ring ◽  
Ring Spectra ◽  
Characteristic 2

ON COMMUTATIVE RING SPECTRA OF CHARACTERISTIC 2

1985 ◽  
Vol 52 (2) ◽  
pp. 471-479 ◽  
Author(s):  
A V Pazhitnov ◽  
Yu B Rudyak
Keyword(s):  
Commutative Ring ◽  
Ring Spectra ◽  
Characteristic 2

scholarly journals Moduli spaces of commutative ring spectra

2004 ◽  
pp. 151-200 ◽  
Author(s):  
P. G. Goerss ◽  
M. J. Hopkins
Keyword(s):  
Commutative Ring ◽  
Moduli Spaces ◽  
Ring Spectra

scholarly journals Brauer groups and Galois cohomology of commutative ring spectra

2021 ◽  
Vol 157 (6) ◽  
pp. 1211-1264
Author(s):  
David Gepner ◽  
Tyler Lawson
Keyword(s):  
Fixed Point ◽  
Spectral Sequence ◽  
Commutative Ring ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Obstruction Theory ◽  
Azumaya Algebras ◽  
Ring Spectra ◽  
K Theory ◽  
Coefficient Ring

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.

scholarly journals SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA

2018 ◽  
Vol 18 (4) ◽  
pp. 707-758 ◽  
Author(s):  
Dmitri Pavlov ◽  
Jakob Scholbach
Keyword(s):  
Algebraic Geometry ◽  
Commutative Ring ◽  
Model Category ◽  
Obstruction Theory ◽  
Model Structure ◽  
Simplicial Sets ◽  
Ring Spectra ◽  
Symmetric Spectra ◽  
Derived Algebraic Geometry ◽  
Quillen Equivalent

This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

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