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 [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

In this paper we show that a conjecture of Voevodsky about the slices of the motivic cobordism spectrum implies a statement about the slices of motivic Landweber spectra. Over perfect fields these slices are given by the coefficients of the corresponding topological Landweber spectrum and the motivic Eilenberg MacLane spectrum. We also prove a cohomological version of Landweber exactness which applies to the compact objects of the stable motivic homotopy category.

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Using the ‘slice filtration’, defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology and étale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their $E_2$-terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.