A new basis for the complex $K$-theory cooperations algebra

TL;DR

This paper introduces a new p-local basis for the complex K-theory cooperations algebra using Adams splitting and Hazewinkel generators, refining the classical basis for p=2.

Contribution

It provides a novel p-local basis for KU_0ku, enhancing the understanding of the algebra's structure through Adams splitting and known generator formulas.

Findings

New p-local basis for KU_0ku_{(p)} established

For p=2, the basis matches the classical basis modulo higher Adams filtration

Uses Adams splitting and Hazewinkel generators to derive the basis

Abstract

A classical theorem of Adams, Harris, and Switzer states that the 0th grading of complex $K$-theory cooperations, $KU_0ku$ is isomorphic to the space of numerical polynomials. The space of numerical polynomials has a basis provided by the binomial coefficient polynomials, which gives a basis of $KU_0ku$. In this paper, we produce a new $p$-local basis for $KU_0ku_{(p)}$ using the Adams splitting. This basis is established by using well known formulas for the Hazewinkel generators. For $p=2$, we show that this new basis coincides with the classical basis modulo higher Adams filtration.