nLab periodic ring spectrum

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).

Examples

even 2-periodic:

• KU

• MP

• elliptic spectrum

• 2-periodic Morava K-theory

• Morava E-theory

• 2-periodic sphere spectrum

References

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

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001, DOI 10.1007/s002220100175

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

• Mark Hovey, Neil Strickland, theorem 2.8 of Morava K-theories and localisation Mem. Amer. Math. Soc., 139(666):viii+100, 1999.

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 , Lecture 18 Even periodic cohomology theories (pdf)

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)