On the ring of cooperations for real hermitian K-theory

TL;DR

This paper provides a detailed algebraic description of the ring of cooperations in real motivic Hermitian K-theory, utilizing spectral sequences and proving a splitting result for symplectic K-theory.

Contribution

It offers a full description of the cooperations ring in real Hermitian K-theory and introduces a novel spectral sequence analysis approach.

Findings

The $E_2$-page of the motivic Adams spectral sequence collapses.

The ring of cooperations is described via Brown--Gitler comodules.

A splitting result for symplectic K-theory over fields of characteristic not two is proved.

Abstract

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations $\pi^\mathbb{R}_{**}(\text{kq} \otimes \text{kq})$ in the stable motivic homotopy category $\text{SH}(\mathbb{R})$, giving a full description in terms of Brown--Gitler comodules. To do this, we decompose the $E_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the $E_2$-page is accomplished by a series of algebraic Atiyah--Hirzebruch spectral sequences which converge to the summands of the $E_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory ksp over any base field of characteristic not two.