2019/10/14 by Li, Guchuan, Lorman, Vitaly, Quigley, J. D.
#55N25 #55P42 #55P60 #55P91 #55P92 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1910.06191
We study transchromatic phenomena for the Tate construction of Real oriented cohomology theories. First, we show that after suitable completion, the Tate construction with respect to a trivial ℤ/2-action on height n Real Johnson--Wilson theory splits into a wedge of height n-1 Real Johnson--Wilson theories. This is the first example of Tate blueshift at all chromatic heights outside of the complex oriented setting. Second, we prove that the Tate construction with respect to a trivial finite group action on Real Morava K-theory vanishes, refining a classical Tate vanishing result of Greenlees--Sadofsky. In the course of proving these results, we develop some ideas in equivariant chromatic homotopy theory (e.g., completions of module spectra over Real cobordism, C2-equivariant chromatic Bousfield localizations) and apply the parametrized Tate construction.