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On the ring of cooperations for real Hermitian K-theory

Jackson Morris

Abstract

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations $π^\mathbb{R}_{**}(\text{kq} \otimes \text{kq})$ 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.

On the ring of cooperations for real Hermitian K-theory

Abstract

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations in the stable motivic homotopy category , giving a full description in terms of Brown--Gitler comodules. To do this, we decompose the -page of the motivic Adams spectral sequence and show that it must collapse. The description of the -page is accomplished by a series of algebraic Atiyah--Hirzebruch spectral sequences which converge to the summands of the -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.

Paper Structure

This paper contains 38 sections, 47 theorems, 249 equations, 38 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Future directions
  5. Organization
  6. Notation and Conventions
  7. Acknowledgments
  8. Preliminaries
  9. Stable motivic homotopy theory
  10. The $\textup{kq}$-resolution
  11. The dual motivic Steenrod algebra
  12. Motivic Brown--Gitler comodules
  13. A splitting of symplectic K-theory
  14. Very effective slice filtration
  15. A splitting of symplectic K-theory
  16. ...and 23 more sections

Key Result

Theorem A

The $\textup{mASS}^{\mathbb{R}}(\textup{kq} \otimes \textup{kq})$ has signature and collapses on the $\textup{E}_2$-page. Here $B_0^{\mathbb{R}}(k)$ denotes the $k^{th}$ integral motivic Brown--Gitler comodule. We describe the $\textup{E}_2=\textup{E}_\infty$-page, modulo $v_1$-torsion, as a module over $\pi_{**}^{\mathbb{R}}\textup{(kq)}.$

Figures (38)

  • Figure 2.1: The motivic Brown--Gitler comodule $B_0^F(1)$
  • Figure 4.1: A heuristic for the aAHSS$(B_0^F(1))$.
  • Figure 5.1: The $\textup{E}_1$-page of the $\textbf{aAHSS}(B_0^\mathbb{C}(1))$
  • Figure 5.2: The $\textup{E}_2$-page of the aAHSS($B_0^\mathbb{C}(1)$)
  • Figure 5.3: The $\textup{E}_3=\textup{E}_\infty$-page of the aAHSS($B_0^\mathbb{C}(1))$ with hidden extensions.
  • ...and 33 more figures

Theorems & Definitions (102)

  • Theorem A: \ref{['e2 mass']}, \ref{['main']}
  • Theorem B: \ref{['prop:n-lineE2']}, \ref{['thm:n-lineDifs']}
  • Theorem C: \ref{['R-ksp']}
  • Theorem D: \ref{['ksp hz1']}
  • Theorem 2.1: CQ21
  • Remark 2.1
  • Theorem 2.2: ARO20
  • Corollary 2.3: CQ21
  • Remark 2.2
  • Theorem 2.4: Voereduced
  • ...and 92 more