Holmstrom Slice filtration

Voevodsky’s Nordfjordeid lecture - supernice (Voevodsky folder). Working over a field unfortunately. Basic constructions, of SH etc. Brief discussion of Thom spaces and homotopy purity. Cohomology theories: the motivic EM spectrum, KGL, MGL, claim that the notions of orientation and FGLs have direct analogs for P1-spectra. The slice filtration (great intro), update on Open problems paper. The zero-th slice of the unit spectrum is HZ, this is known for fields of char zero. Brief discussion of AHSS. Appendix on the Nisnevich topology, Nisnevich descent, and model structures.

Check writings by Voevodsky, Levine, Huber. Relation to coniveau stuff?

Slice filtration on motives and the Hodge conjecture, by Annette Huber

Open problems paper of Voevodsky

Voevodsky: Motives over simplicial schemes, contains some material. He says that Lemma 5.12 (p 22) about Homs between various truncations (vanishing etc) holds in the triang cat of motives but not in SH, and this is the major difference between the slice filtration in the two settings. A counterexample in the SH setting is the Hopf map, see p 22 for more. Probably this also affects the adjointness properties of the truncations.

arXiv:1002.0317 On the Functoriality of the Slice Filtration from arXiv Front: math.AT by Pablo Pelaez Let $\mathbf{HZ}_{X}\otimes \mathbb Qk$ 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 [MR991991] extending the result of Levine [MR2365658], and also the zero slice of the sphere spectrum extending the result of Levine [MR2365658] and Voevodsky [MR2101286]. 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 [mixedmotives], 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 \cite[conjectures 1, 7, 10, 11]{MR1977582}.

nLab page on Slice filtration