On the ring of cooperations for real hermitian K-theory

2025/06/20 by Jackson Morris, Morris, Jackson · 2 voices · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AG #math.AT #math.KT

paper · pdf · doi:10.48550/arxiv.2506.16672

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations π^ℝ_**(kq ⊗ kq) in the stable motivic homotopy category SH(ℝ), giving a full description in terms of Brown--Gitler comodules. To do this, we decompose the E₂-page of the motivic Adams spectral sequence and show that it must collapse. The description of the E₂-page is accomplished by a series of algebraic Atiyah--Hirzebruch spectral sequences which converge to the summands of the E₂-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.

Cited by

• Rings of cooperations for Hermitian K-theory over finite fields

Discussions

• My grad student @jacksonmorris.bsky.social just posted his first preprint, a 50-page analysis of coöperations in Hermitian K-theory over the real numbers! arxiv.org/abs/2506.16672 🧮 [bsky, 7 points, 1 comments]
• Paper up on the Arxiv! I’m so excited to share this project with the world. This is the first part of my thesis and my first real piece of research, and there are so many feelings that it’s hard to pu [bsky, 3 points, 0 comments]

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