On the Functoriality of the Slice Filtration

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{\'e}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.