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Authors
Birgit Richter
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In this survey paper on commutative ring spectra we present some basic features of commutative ring spectra and discuss model category structures. As a first interesting class of examples of such ring spectra we focus on (commutative) algebra spectra over commutative Eilenberg-MacLane ring spectra. We present two constructions that yield commutative ring spectra: Thom spectra associated to infinite loop maps and Segals construction starting with bipermutative categories. We define topological Hochschild homology, some of its variants, and topological Andre-Quillen homology. Obstruction theory for commutative structures on ring spectra is described in t
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In this paper we develop methods for classifying Baker-Richter-Szymiks 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 Luries 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.
We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null. If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.
We describe a special instance of the Goerss-Hopkins obstruction theory, due to Senger, for calculating the moduli of $E_infty$ ring spectra with given mod-$p$ homology. In particular, for the $2$-primary Brown-Peterson spectrum we give a chain complex that calculates the first obstruction groups, locate the first potential genuine obstructions, and discuss how some of the obstruction classes can be interpreted in terms of secondary operations.
Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological Andre-Quillen homology. In the classical algebraic context it is important to distinguish between tame and wild ramification. Noethers theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the trace map. We transfer the latter fact to ring spectra and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. We study ramification in examples in the context of topological K-theory and topological modular forms.
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