The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

Abstract:

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 $TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z$ in terms of units in the $p$-cyclotomic tower of ${\mathbb {Q}}_{p}$.