# nLab periodic ring spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

A periodic ring spectrum is a ring spectrum/A-∞ ring which represents a periodic cohomology theory.

A common case are the even periodic or 2-periodic ring spectra, in particular those representing even cohomology theories.

## Properties

### Periodicity of the $\infty$-category of $\infty$-modules

For $E$ an E-∞ ring representing a periodic ring spectrum, double suspension/looping on any $E$-∞-module $N$ is equivalent to the identity

$\Omega^2 N \simeq N \simeq \Sigma^2 N \,.$

This equivalence ought to be coherent to yield a $\mathbb{Z}/2\mathbb{Z}$ ∞-action on the (∞,1)-category of (∞,1)-modules $E Mod$ (MO discussion).

### Landweber exact functor theorem

There is an analogue of the Landweber exact functor theorem for even 2-periodic cohomology theories, with MU replaced by MP (Hovey-Strickland 99, theorem 2.8, Lurie lecture 18, prop. 11).

even 2-periodic:

## References

The concept of even 2-periodic multiplicative cohomology theories originates with

The analogue of the Landweber exact functor theorem for even 2-periodic cohomology is discussed in

The $\mathbb{Z}/2$-graded formalism (supercommutative superalgebra) for modules over $E_\infty$-algebras over an even periodic ring spectrum:

• Charles Rezk, The congruence criterion for power operations in Morava E-theory, Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.327-379 (arXiv:0902.2499)

See also

• Akhil Mathew, Lennart Meier, section 2.1 of Affineness and chromatic homotopy theory, J. Topol. 8 (2015), no. 2, 476–528 (arXiv:1311.0514)

Last revised on September 10, 2021 at 05:18:04. See the history of this page for a list of all contributions to it.