莱夫谢茨动机的零化子
The annihilator of the Lefschetz motive
摘要：
本文研究了谱$K(\mathcal{V}_k)$，使得$π_0 K(\mathcal{V}_k)$是簇的Grothendieck环，且高阶同伦群包含更多关于簇几何的几何信息.我们使用这个谱的拓扑结构分析$K_0[\mathcal{V}_k]$的结构，并表明，类的核心乘以$[\mathbb{A}^1]$可以始终表示为$[X]-[Y]$，其中$X$和$Y$是品种，$[X] \neq [Y]$，$X\times \mathbb{A}^1$和$Y\times \mathbb{A}^1$不是分段同构，但$[X\times \mathbb{A}^1] =[Y\times \mathbb{A}^1]$在$K_0[\mathcal{V}_k]$。在此过程中，我们给出了Larsen-Lunts关于$K_0[\mathcal{V}_k]/([\mathbb{A}^1])$结构的结果的新证明.
英文摘要：
In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $π_0 K(\mathcal{V}_k)$ 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 $[X] \neq [Y]$, $X\times \mathbb{A}^1$ 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 new proofs of the result of Larsen--Lunts on the structure on $K_0[\mathcal{V}_k]/([\mathbb{A}^1])$.
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