Power operations in $K$-theory completed at a prime

TL;DR

This paper analyzes the structure of power operations in p-completed K-theory, providing detailed descriptions of their actions on cooperation algebras at specific primes, enhancing understanding of algebraic operations in stable homotopy theory.

Contribution

It offers a detailed description of power operations on p-completed cooperation algebras in K-theory at primes p and 2, advancing the algebraic understanding of these operations.

Findings

Explicit descriptions of power operations on K-theory cooperation algebras.

Clarification of algebraic structures at prime p and 2.

Enhanced tools for computations in stable homotopy theory.

Abstract

We describe the action of power operations on the $p$-completed cooperation algebras $K^\vee_0 K = K_0(K)\sphat_p$ for $K$-theory at a prime~$p$, and $K^\vee_0 KO = K_0(KO)\sphat_2$.