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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The $ K$-theory spectrum of varieties


Author: Jonathan A. Campbell
Journal: Trans. Amer. Math. Soc. 371 (2019), 7845-7884
MSC (2010): Primary 19D99
DOI: https://doi.org/10.1090/tran/7648
Published electronically: February 14, 2019
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Abstract: We produce an $ E_\infty $-ring spectrum $ K(\mathbf {Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $ k$) $ K_0 (\mathbf {Var}_{/k})$. This is achieved by slightly modifying Waldhausen categories and the Waldhausen $ S_\bullet $-construction. As an application, we 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})$.


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Additional Information

Jonathan A. Campbell
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240
Email: j.campbell@vanderbilt.edu

DOI: https://doi.org/10.1090/tran/7648
Keywords: Grothendick ring of varieties, K-theory, $S_\bullet$-construction, motivic measure
Received by editor(s): January 9, 2017
Received by editor(s) in revised form: February 14, 2018, February 28, 2018, and March 23, 2018
Published electronically: February 14, 2019
Article copyright: © Copyright 2019 American Mathematical Society