- 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.

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