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periodic ring spectrum

Contents

Context

Stable Homotopy theory

Higher 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 EE an E-∞ ring representing a periodic ring spectrum, double suspension/looping on any EE-∞-module NN is equivalent to the identity

Ω 2NNΣ 2N. \Omega^2 N \simeq N \simeq \Sigma^2 N \,.

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

Examples

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 /2\mathbb{Z}/2-graded formalism (supercommutative superalgebra) for modules over E 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 May 29, 2017 at 02:53:08. See the history of this page for a list of all contributions to it.