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

Let p ? Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for cyclotomic trace of sphere spectrum alttext="double-struck S"> mathvariant="double-struck">S encoding="application/x-tex">{\mathbb {S}} admits “eigensplitting” generalizes known splittings on alttext="upper K"> K encoding="application/x-tex">K -theory and T C"> T C encoding="application/x-tex">TC . identify summands in as covers Z Subscript p"> {Z}}_{p} -Anderson duals K left-parenthesis 1 right-parenthesis"> ( 1 stretchy="false">) encoding="application/x-tex">K(1) -localized algebraic Analogous results hold ring where we prove is self-dual duality, with duality permuting by alttext="i right-arrow from bar p minus i"> i stretchy="false">? ? encoding="application/x-tex">TC({\mathbb {Z}})^{\wedge }_{p}\simeq \vee \Sigma j’\vee Z terms units alttext="p"> encoding="application/x-tex">p -cyclotomic tower Q mathvariant="double-struck">Q {Q}}_{p}