symmetric monoidal (∞,1)-category of spectra
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.
For $E$ an E-∞ ring representing a periodic ring spectrum, double suspension/looping on any $E$-∞-module $N$ is equivalent to the identity
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).
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:
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
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:
See also
Last revised on September 10, 2021 at 09:18:04. See the history of this page for a list of all contributions to it.