On the ring of cooperations for real hermitian K-theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909040722837504 |
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| author | Morris, Jackson |
| author_facet | Morris, Jackson |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the ring of cooperations for real hermitian K-theory Morris, Jackson Algebraic Topology Algebraic Geometry K-Theory and Homology 14F42, 55Q10, 55Q51, 55T15, 19G38 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. |
| title | On the ring of cooperations for real hermitian K-theory |
| topic | Algebraic Topology Algebraic Geometry K-Theory and Homology 14F42, 55Q10, 55Q51, 55T15, 19G38 |
| url | https://arxiv.org/abs/2506.16672 |