Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866909063156072448 |
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| author | Carmody, Daniel |
| author_facet | Carmody, Daniel |
| contents | We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $\frac{1}{2} \in R$; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_\infty$ motivic ring spectrum $\mathbf{KR}^{\mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that $\mathbf{KR}^{\mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_09124 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution Carmody, Daniel K-Theory and Homology Algebraic Geometry Algebraic Topology We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $\frac{1}{2} \in R$; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_\infty$ motivic ring spectrum $\mathbf{KR}^{\mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that $\mathbf{KR}^{\mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence. |
| title | Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2009.09124 |