Tate blueshift and vanishing for Real oriented cohomology

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 $\mathbb{Z}/2$-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, $C_2$-equivariant chromatic Bousfield localizations) and apply the parametrized Tate construction.

arXiv comment38 pages. v2: Revised exposition and expanded some proofs. v3: Updated funding sources and bibliography. v4: Revised intro, clarified some arguments. Comments welcome!

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Cited by in corpus (3)

• Real motivic and $C_2$-equivariant Mahowald invariants
• On the parametrized Tate construction
• Odd primary analogs of Real orientations