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Tate blueshift and vanishing for Real oriented cohomology

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 Added by J.D. Quigley
 Publication date 2019
  fields
and research's language is English




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Ando, Morava, and Sadofsky showed that the Tate construction for a trivial $mathbb{Z}/p$-action decreases the chromatic height of Johnson-Wilson theory, and Greenlees and Sadofsky proved that the Tate construction for a trivial finite group action vanishes on Morava K-theory. We prove $C_2$-equivariant enrichments of these results using the parametrized Tate construction. The $C_2$-fixed points of our results produce new blueshift and vanishing results for Real Johnson-Wilson theories $ER(n)$ and Real Morava $K$-theories $KR(n)$, respectively, for all $n$. In particular, our blueshift results generalize Greenlees and Mays Tate splitting of $KO$ to all chromatic heights.



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We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{mathbb{R}}$.
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We take a direct approach to computing the orbits for the action of the automorphism group $mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{mathbb{G}_2} cong mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $pgeq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.
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