Ausoni-Rognes猜想的多个基本情形下某些对象的构造
A construction of some objects in many base cases of an Ausoni-Rognes conjecture
摘要：
设$p$是素数，$n \geq 1$，$K(n)$是$n$的MoravaPDMATHABMQ-理论谱，$\mathbb{G}_n$是扩展的Morava稳定群，$K(A)$是交换$S$ -代数$A$的代数$K$ -理论谱.对于$n+1$型复形$V_n$，Ausoni和Rognes证明了（a）从$K(n)$ -局部球面到Lubin-Tate谱的单位映射$i_n: L_{K(n)}(S^0) \to E_n$诱导出一个弱等价的映射\[K(L_{K(n)}(S^0)) \wedge v_{n+1}^{-1}V_n \to (K(E_n))^{h\mathbb{G}_n} \wedge v_{n+1}^{-1}V_n\]，其中（b）由于$\mathbb{G}_n$是profinite的，所以$(K(E_n))^{h\mathbb{G}_n}$表示连续同伦不动点谱，以及（c）上述映射的目标的$π_\ast(-)$是同伦不动点谱序列的邻接。对于$n = 1$、$p \geq 5$和$V_1 = V(1)$，我们给出了实现上述映射的一种方法和（c），通过证明$i_1$诱导一个映射\[K(L_{K(1)}(S^0)) \wedge v_{2}^{-1}V_1 \to (K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1},\]，其中该映射的目标是一个连续同伦不动点谱，具有一个相关的同伦不动点谱序列。证明了当$\mathbb{G}_1$为离散群时，存在一个等价的同伦不动点\[(K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1} \simeq (K(E_1))^{\widetilde{h}\mathbb{G}_1} \wedge v_2^{-1}V_1,\]，其中$(K(E_1))^{\widetilde{h}\mathbb{G}_1}$为同伦不动点.
英文摘要：
Let $p$ be a prime, $n \geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $\mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) \to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map \[K(L_{K(n)}(S^0)) \wedge v_{n+1}^{-1}V_n \to (K(E_n))^{h\mathbb{G}_n} \wedge v_{n+1}^{-1}V_n\] that is a weak equivalence, where (b) since $\mathbb{G}_n$ is profinite, $(K(E_n))^{h\mathbb{G}_n}$ denotes a continuous homotopy fixed point spectrum, and (c) $π_\ast(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n = 1$, $p \geq 5$, and $V_1 = V(1)$, we give a way to realize the above map and (c), by proving that $i_1$ induces a map \[K(L_{K(1)}(S^0)) \wedge v_{2}^{-1}V_1 \to (K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1},\] where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence \[(K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1} \simeq (K(E_1))^{\widetilde{h}\mathbb{G}_1} \wedge v_2^{-1}V_1,\] where $(K(E_1))^{\widetilde{h}\mathbb{G}_1}$ is the homotopy fixed points with $\mathbb{G}_1$ regarded as a discrete group.
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