Tate blueshift and vanishing for Real oriented cohomology

DOI10.1016/J.AIM.2022.108780zbMATH Open1528.55003arXiv1910.06191MaRDI QIDQ6327153

J. D. Quigley, Vitaly Lorman, Guchuan Li

Publication date: 14 October 2019

Abstract: 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 $mathbbZ/2$-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, $C_2$-equivariant chromatic Bousfield localizations) and apply the parametrized Tate construction.

Mathematics Subject Classification ID

Localization and completion in homotopy theory (55P60) Stable homotopy theory, spectra (55P42) Equivariant homotopy theory in algebraic topology (55P91) Homology with local coefficients, equivariant cohomology (55N25) Relations between equivariant and nonequivariant homotopy theory in algebraic topology (55P92)

This page was built for publication: Tate blueshift and vanishing for Real oriented cohomology