论文标题
球形光谱的循环迹线的纤维的特征插图
The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum
论文作者
论文摘要
令$ p \ in \ mathbb z $成为一个奇怪的素数。我们表明，球形频谱$ \ mathbb s $的循环痕迹的光纤序列承认了一个“ eigensplitting”，它概括了$ k $ - 理论和$ tc $的已知分组。我们将光纤中的汇总确定为$ \ mathbb z_ {p} $ - $ k（1）$ - 本地化代数$ k $的封面。 ring $ \ mathbb z $的类似结果可以证明$ k（1）$ - 局部纤维序列是$ \ mathbb z_ {p} $ - 安德森二重性的自二，而双重性将$ i \ i \ i \ mapsto p-i $ p-i $（Indexexed mod $ p-1 $ p-1 $ $ p-1 $）定为二元。我们解释了汇总的固有特征，我们称$ z $在$ tc（\ mathbb z）中称为$ z $^{\ wedge} _ {p} \ simeq j \ veeσj'\ veeσj'\ vee z $在$ p $ p $ cycyclotomic tower of $ \ m m i \ mathbb q_ p}中。
Let $p\in \mathbb Z$ be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum $\mathbb S$ admits an "eigensplitting" that generalizes known splittings on $K$-theory and $TC$. We identify the summands in the fiber as the covers of $\mathbb Z_{p}$-Anderson duals of summands in the $K(1)$-localized algebraic $K$-theory of $\mathbb Z$. Analogous results hold for the ring $\mathbb Z$ where we prove that the $K(1)$-localized fiber sequence is self-dual for $\mathbb Z_{p}$-Anderson duality, with the duality permuting the summands by $i\mapsto p-i$ (indexed mod $p-1$). We explain an intrinsic characterization of the summand we call $Z$ in the splitting $TC(\mathbb Z)^{\wedge}_{p}\simeq j \vee Σj'\vee Z$ in terms of units in the $p$-cyclotomic tower of $\mathbb Q_{p}$.
