# Rings, Modules, and Algebras in Stable Homotopy Theory

@inproceedings{Elmendorf2007RingsMA, title={Rings, Modules, and Algebras in Stable Homotopy Theory}, author={Anthony Elmendorf and Michael Cole}, year={2007} }

Introduction Prologue: the category of ${\mathbb L}$-spectra Structured ring and module spectra The homotopy theory of $R$-modules The algebraic theory of $R$-modules $R$-ring spectra and the specialization to $MU$ Algebraic $K$-theory of $S$-algebras $R$-algebras and topological model categories Bousfield localizations of $R$-modules and algebras Topological Hochschild homology and cohomology Some basic constructions on spectra Spaces of linear isometries and technical theorems The monadic bar…

## 604 Citations

Global model structures for $\ast$-modules

- MathematicsHomology, Homotopy and Applications
- 2019

We extend Schwede's work on the unstable global homotopy theory of orthogonal spaces and $\mathcal{L}$-spaces to the category of $*$-modules (i.e., unstable $S$-modules). We prove a theorem which…

$G_\infty$-ring spectra and Moore spectra for $\beta$-rings

- Mathematics
- 2020

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their equivariant…

GENERALIZED THOM SPECTRA AND THEIR TOPOLOGICAL HOCHSCHILD HOMOLOGY

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2017

We develop a theory of $R$ -module Thom spectra for a commutative symmetric ring spectrum $R$ and we analyze their multiplicative properties. As an interesting source of examples, we show that $R$…

Quillen-Segal algebras and Stable homotopy theory.

- Mathematics
- 2018

Let $\mathscr{M}$ be a monoidal model category that is also combinatorial and left proper. If $\mathscr{O}$ is a monad, operad, properad, or a PROP; following Segal's ideas we develop a theory of…

An introduction to higher categorical algebra

- Mathematics
- 2019

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing…

The Homology of Connective Morava $E$-theory with coefficients in $\mathbb{F}_p$

- Mathematics
- 2017

Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;\mathbb{F}_p)$ for any prime $p$ and $n \leq 4$ up to possible…

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

- MathematicsAlgebraic & Geometric Topology
- 2018

We give a description of the factorization homology and $E_n$ topological Hochschild cohomology of Thom spectra arising from $n$-fold loop maps $f: A \to BO$, where $A = \Omega^n X$ is an $n$-fold…

Derived representation theory of Lie algebras and stable homotopy categorification of sl

- MathematicsAdvances in Mathematics
- 2019

## References

SHOWING 1-10 OF 72 REFERENCES

Basic constructions in the $K$-theory of homotopy ring spaces

- Mathematics
- 1994

Using the language of category theory and universal algebra we formalize the passage from the permutative category of finitely generated free Rmodules to the algebraic K-theory KR of R and thus make…

TOWARD AN ALGEBRAIC CLASSIFICATION OF MODULE SPECTRA

- Mathematics
- 1997

The category of modules over an S-algebra (A∞ or E∞ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the S-algebra in question form a…

Commutative algebra in stable homotopy theory and a completion theorem

- Mathematics
- 1994

We construct a category of spectra that has all limits and col- imits and also has a strictly associative and commutative smash product. This provides the ground category for a new theory of…

Spectra of derived module homomorphisms

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1987

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a…

The cyclotomic trace and algebraic K-theory of spaces

- Mathematics
- 1993

The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and…

H Ring Spectra and Their Applications

- Mathematics, Chemistry
- 1986

Extended powers and H? ring spectra.- Miscellaneous applications in stable homotopy theory.- Homology operations for H? and Hn ring spectra.- The homotopy theory of H? ring spectra.- The homotopy…