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Register a new userThe annihilator of the Lefschetz motive
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Authors
Inna Zakharevich
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In this paper 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 show that classes in the kernel of multiplication by $[mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are varieties such that $[X] eq [Y]$, $Xtimes mathbb{A}^1$ and $Ytimes mathbb{A}^1$ are not piecewise isomorphic, but $[Xtimes mathbb{A}^1] =[Ytimes mathbb{A}^1]$ in $K_0[mathcal{V}_k]$. Along the way we present new proofs of the result of Larsen--Lunts on the structure on $K_0[mathcal{V}_k]/([mathbb{A}^1])$.
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