On the ring of cooperations for real hermitian K-theory

Jackson Morris

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arXiv · 2506.16672

On the ring of cooperations for real hermitian K-theory

Jackson Morris

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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.

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Jackson Morris. 2026-05-13. On the ring of cooperations for real hermitian K-theory. https://arxiv.org/abs/2506.16672

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