Duke Mathematical Journal

The annihilator of the Lefschetz motive

Inna Zakharevich

Abstract

In this article, we study a spectrum $K(\mathcal{V}_{k})$ such that $\pi_{0}K(\mathcal{V}_{k})$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_{0}[\mathcal{V}_{k}]$ and to show that classes in the kernel of multiplication by $[\mathbb{A}^{1}]$ can always be represented as $[X]-[Y]$, where $[X]\neq[Y]$, $X\times\mathbb{A}^{1}$, and $Y\times\mathbb{A}^{1}$ are not piecewise-isomorphic, but $[X\times\mathbb{A}^{1}]=[Y\times\mathbb{A}^{1}]$ in $K_{0}[\mathcal{V}_{k}]$. Along the way, we present a new proof of the result of Larsen–Lunts on the structure on $K_{0}[\mathcal{V}_{k}]/([\mathbb{A}^{1}])$.

Article information

Source
Duke Math. J., Volume 166, Number 11 (2017), 1989-2022.

Dates
Received: 29 September 2015
Revised: 20 November 2016
First available in Project Euclid: 28 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1493344842

Digital Object Identifier
doi:10.1215/00127094-0000016X

Mathematical Reviews number (MathSciNet)
MR3694563

Zentralblatt MATH identifier
06775425

Subjects
Primary: 19E99: None of the above, but in this section
Secondary: 14E99: None of the above, but in this section

Citation

Zakharevich, Inna. The annihilator of the Lefschetz motive. Duke Math. J. 166 (2017), no. 11, 1989--2022. doi:10.1215/00127094-0000016X. https://projecteuclid.org/euclid.dmj/1493344842

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