论切片过滤的实用性
On the Functoriality of the Slice Filtration
摘要：
设$k$是奇点的分解的域，$X$是具有结构映射$g$的有限型分离$k$ -概型.我们证明了motivic稳定同伦范畴中的切片过滤与沿$g$的拉回交换.限制领域进一步的情况下，特征零，我们能够计算切片的韦伯同伦不变量$K$ -理论扩展Levine的结果，也零切片的球谱扩展Levine和Voevodsky的结果。我们还证明了球谱的零切片是一个严格凝聚环谱$\mathbf{HZ}_{X}^{\slicefilt}$，它在拉回下是稳定的，并且所有切片都具有$\mathbf{HZ}_{X}^{\slicefilt}$上严格模的正则结构。如果考虑有理系数，并假设$X$是几何单支的，那么借助于Cisinski和D{é}glise的工作，我们得到球谱的零切片由Voevodsky的有理动机上同调谱$\mathbf{HZ}_{X}\otimes \mathbb Q$给出，并且切片具有转移.这证明了Voevodsky的几个论点。
英文摘要：
Let $k$ be a field with resolution of singularities, and $X$ a separated $k$-scheme of finite type with structure map $g$. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along $g$. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant $K$-theory extending the result of Levine, and also the zero slice of the sphere spectrum extending the result of Levine and Voevodsky. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum $\mathbf{HZ}_{X}^{\slicefilt}$ which is stable under pullback and that all the slices have a canonical structure of strict modules over $\mathbf{HZ}_{X}^{\slicefilt}$. If we consider rational coefficents and assume that $X$ is geometrically unibranch then relying on the work of Cisinski and D{é}glise, we get that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum $\mathbf{HZ}_{X}\otimes \mathbb Q$ and that the slices have transfers. This proves several conjectures of Voevodsky.
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