In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map \(T(n)\rightarrow BP\) of homotopy commutative ring spectra can be rigidified to map of \(E_2\) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

在1980年代，Ravenel引入了光谱X（n）和T（n）的序列，这些序列在证明德维纳茨-霍普金斯-史密斯的幂等定理中起着重要作用。在本文中，我们解决了X（n）的拓扑Hochschild同源性的同伦极限问题，这是本素猜想针对素数阶环组的广义形式。该结果是使用迹线方法计算X（n）的代数K理论的第一步，该方法在精确的意义上近似了球谱的代数K理论。我们解决了T的拓扑Hochschild同源性的同伦极限问题（n）假设同构交换环谱的典范图\（T（n）\ rightarrow BP \）可以被硬化为\（E_2 \）环谱图。我们表明，可以用Atiyah-Hirzebruch光谱序列中的显式类来描述对我们的假设的阻碍。