Tate blueshift and vanishing for Real oriented cohomology

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.

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