Segal K-theory of vector spaces with an automorphism
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909353818193920 |
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| author | Bianchi, Andrea Kranhold, Florian |
| author_facet | Bianchi, Andrea Kranhold, Florian |
| contents | We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of maps from $S^1$ to the disjoint union of classifying spaces $\mathrm{BGL}_d(\mathbb F)$, in terms of the $K$-theory of finite field extensions of $\mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field $\mathbb F$. We also discuss the topological cases of $\mathbb F =\mathbb C,\mathbb R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Segal K-theory of vector spaces with an automorphism Bianchi, Andrea Kranhold, Florian K-Theory and Homology Algebraic Topology 19D23, 19D06, 18E10, 15A20, 12F10, 55P42, 55947 We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of maps from $S^1$ to the disjoint union of classifying spaces $\mathrm{BGL}_d(\mathbb F)$, in terms of the $K$-theory of finite field extensions of $\mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field $\mathbb F$. We also discuss the topological cases of $\mathbb F =\mathbb C,\mathbb R$. |
| title | Segal K-theory of vector spaces with an automorphism |
| topic | K-Theory and Homology Algebraic Topology 19D23, 19D06, 18E10, 15A20, 12F10, 55P42, 55947 |
| url | https://arxiv.org/abs/2407.01482 |