- CiteKey: “Cam17” - Type: report - Title: “The K-Theory Spectrum of Varieties,” - Author: “Campbell, Jonathan A.;” - Publisher: “arXiv,” - Year: 2017 - Collections: “Syllabus; Talbot 2022,” - Keywords: “Mathematics - Algebraic Geometry”; “Mathematics - Algebraic Topology”; “Mathematics - K-Theory and Homology”
The K-Theory Spectrum of Varieties
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- URL: http://arxiv.org/abs/1505.03136
- URI: http://zotero.org/users/1049732/items/P4RTGA3V
- Open in Zotero: Zotero
Abstract
Using a construction closely related to Waldhausen’s \(S_\bullet\)-construction, we produce a spectrum \(K(\mathbf{Var}_{/k})\) whose components model the Grothendieck ring of varieties (over a field \(k\)) \(K_0 (\mathbf{Var}_{/k})\). We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausen’s \(K\)-theory of spaces \(A(\ast)\) and to \(K(\mathbf{Q})\). We end with a conjecture relating \(K(\mathbf{Var}_{/k})\) and the doubly-iterated \(K\)-theory of the sphere spectrum.