EVERY K(n)-LOCAL SPECTRUM IS THE HOMOTOPY FIXED POINTS OF ITS MORAVA MODULE
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چکیده
Let n ≥ 1 and let p be any prime. Also, let En be the LubinTate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization LK(n)(X) is equivalent to the homotopy fixed point spectrum (LK(n)(En ∧ X))hGn , which is formed with respect to the continuous action of Gn on LK(n)(En ∧ X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π∗(LK(n)(X)) is isomorphic to the descent spectral sequence that abuts to π∗((LK(n)(En ∧X))hGn ).
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