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 slicefiltration in the motivic stable homotopy category commutes with pullback along$g$. Restricting the field further to the case of characteristic zero, we areable to compute the slices of Weibel's homotopy invariant $K$-theory extendingthe result of Levine, and also the zero slice of the sphere spectrum extendingthe result of Levine and Voevodsky. We also show that the zero slice of thesphere spectrum is a strict cofibrant ring spectrum$\mathbf{HZ}_{X}^{\slicefilt}$ which is stable under pullback and that all theslices have a canonical structure of strict modules over$\mathbf{HZ}_{X}^{\slicefilt}$. If we consider rational coefficents and assumethat $X$ is geometrically unibranch then relying on the work of Cisinski andD{\'e}glise, we get that the zero slice of the sphere spectrum is given byVoevodsky's rational motivic cohomology spectrum $\mathbf{HZ}_{X}\otimes\mathbb Q$ and that the slices have transfers. This proves several conjecturesof Voevodsky.
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机译：假设$ k $是具有奇异分辨率的字段，而$ X $是具有结构图$ g $的有限类型的单独$ k $方案。我们表明，动机稳定的同伦类类别中的切片过滤随着$ g $的回撤而转换。将场限制在特征零的情况下，我们能够计算扩展Levine结果的Weibel同态不变$ K $理论的切片，以及扩展Levine和Voevodsky结果的球谱的零切片。我们还表明，球面频谱的零切片是严格的共纤环谱$ \ mathbf {HZ} _ {X} ^ {\ slicefilt} $，在回拉下稳定，并且所有切片均具有超过$的严格模块的规范结构\ mathbf {HZ} _ {X} ^ {\ slicefilt} $。如果我们考虑有理系数，并假设$ X $在几何上是单分支的，则依靠Cisinski和D {\'e} glise的工作，我们得到球面谱的零切片是由Voevodsky的有理动机同调谱$ \ mathbf {HZ } _ {X} \ otimes \ mathbb Q $，并且切片具有转移。这证明了沃沃斯基的几个猜想。
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