$K(1)$-local $K$-theory of Azumaya algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910015966674944 |
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| author | Ramzi, Maxime |
| author_facet | Ramzi, Maxime |
| contents | We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a Künneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\neq p$, $\mathbf{Br}(R)[p^\infty]\simeq \mathbb{G}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes\mathbb{S}_{W(\overline{\mathbb F}_p)})[p^\infty]$, where $\mathbb{G}_{\mathrm{pic}}$ is Carmeli's strict Picard spectrum. We prove the same result for $R[\frac{1}{p}]$, when $R$ is $p$-henselian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $K(1)$-local $K$-theory of Azumaya algebras Ramzi, Maxime K-Theory and Homology Algebraic Geometry Algebraic Topology We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a Künneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\neq p$, $\mathbf{Br}(R)[p^\infty]\simeq \mathbb{G}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes\mathbb{S}_{W(\overline{\mathbb F}_p)})[p^\infty]$, where $\mathbb{G}_{\mathrm{pic}}$ is Carmeli's strict Picard spectrum. We prove the same result for $R[\frac{1}{p}]$, when $R$ is $p$-henselian. |
| title | $K(1)$-local $K$-theory of Azumaya algebras |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2509.01516 |