---
- 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

## Meta

- **URL**: <http://arxiv.org/abs/1505.03136>
- **URI**: <http://zotero.org/users/1049732/items/P4RTGA3V>
- **Local File**: [arXiv.org Snapshot](file:///home/zack/Zotero/storage/3WQ9HRVP/1505.html); [Campbell - 2017 - The K-Theory Spectrum of Varieties.pdf](file:///home/zack/Zotero/storage/558MZ2CF/Campbell%20-%202017%20-%20The%20K-Theory%20Spectrum%20of%20Varieties.pdf)
- **Open in Zotero**: [Zotero](zotero://select/library/items/P4RTGA3V)

## 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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## Extracted Annotations