Segal K-theory of vector spaces with an automorphism

[2407.01482] Segal K-theory of vector spaces with an automorphism Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > K-Theory and Homology arXiv:2407.01482 (math) [Submitted on 1 Jul 2024 ( v1 ), last revised 18 Oct 2024 (this version, v2)] Title: Segal K-theory of vector spaces with an automorphism Authors: Andrea Bianchi , Florian Kranhold View a PDF of the paper titled Segal K-theory of vector spaces with an automorphism, by Andrea Bianchi and 1 other authors View PDF HTML (experimental) Abstract: 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 $\mathbb F =\mathbb C,\mathbb R$. Comments: 22 pages; accepted version Subjects: K-Theory and Homology (math.KT) ; Algebraic Topology (math.AT) MSC classes: 19D23, 19D06, 18E10, 15A20, 12F10, 55P42, 55947 Report number: CPH-GEOTOP-DNRF151; MPIM-Bonn-2024 Cite as: arXiv:2407.01482 [math.KT] (or arXiv:2407.01482v2 [math.KT] for this version) https://doi.org/10.48550/arXiv.2407.01482 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1093/imrn/rnaf028 Focus to learn more DOI(s) linking to related resources Submission history From: Florian Kranhold [ view email ] [v1] Mon, 1 Jul 2024 17:12:48 UTC (257 KB) [v2] Fri, 18 Oct 2024 13:57:36 UTC (258 KB) Full-text links: Access Paper: View a PDF of the paper titled Segal K-theory of vector spaces with an automorphism, by Andrea Bianchi and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.KT < prev | next > new | recent | 2024-07 Change to browse by: math math.AT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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