Norms and Hermitian $\mathrm{K}$-Theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911417832046592 |
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| author | Shin, Brian |
| author_facet | Shin, Brian |
| contents | Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we discuss an algebro-geometric analogue of this framework, called the theory of normed algebras in motivic spectra. Specifically, we show that the motivic spectrum $\mathrm{ko}$ representing very effective hermitian $\mathrm{K}$-theory can be equipped with a normed algebra structure, and that the orientation map $\mathrm{MSL} \to \mathrm{ko}$ respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03021 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Norms and Hermitian $\mathrm{K}$-Theory Shin, Brian K-Theory and Homology Algebraic Geometry Algebraic Topology Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we discuss an algebro-geometric analogue of this framework, called the theory of normed algebras in motivic spectra. Specifically, we show that the motivic spectrum $\mathrm{ko}$ representing very effective hermitian $\mathrm{K}$-theory can be equipped with a normed algebra structure, and that the orientation map $\mathrm{MSL} \to \mathrm{ko}$ respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms. |
| title | Norms and Hermitian $\mathrm{K}$-Theory |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2602.03021 |