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.

💡 Research Summary

The paper investigates the behavior of the slice filtration in the motivic stable homotopy category 𝖲ℍ(k) under base‑change, and applies the resulting functoriality to concrete computations of slices for important motivic spectra.

Main Theorem (Pull‑back Compatibility).
Let k be a field admitting resolution of singularities and let g : X → Spec k be the structure morphism of a separated k‑scheme of finite type. For any motivic spectrum E∈𝖲ℍ(k) and any integer n, the n‑th slice satisfies a natural isomorphism
 sₙ(g⁎E) ≅ g⁎sₙ(E).
The proof proceeds by first showing that the slice filtration is compatible with the weight structure on 𝖲ℍ(k). Since the weight structure is preserved by the exact pull‑back functor g⁎ (thanks to resolution of singularities and Nisnevich descent), the slices, which are defined as truncations with respect to this weight structure, commute with g⁎. The argument uses the full‑faithfulness of the inclusion of effective motives, the stability of T‑suspension, and the fact that the slice tower is functorial in the motivic model category.

Application I – Weibel’s Homotopy‑Invariant K‑Theory (KH).
Assuming char k = 0, the authors apply the pull‑back compatibility to the homotopy‑invariant K‑theory spectrum KH. Extending Levine’s result from regular schemes, they obtain an explicit description of the slices:
 sₙ(KH_X) ≅ Σ_Tⁿ ℍℤ_X ⊗ K_{‑n}(X),
where Σ_T denotes T‑suspension, ℍℤ_X is the motivic cohomology spectrum with integral coefficients over X, and K_{‑n}(X) denotes the negative K‑groups of X. This identification shows that each slice of KH is a shift of motivic cohomology twisted by the corresponding negative K‑group, thereby linking homotopy‑invariant K‑theory directly to classical K‑theory via the slice filtration.

Application II – The Zero Slice of the Sphere Spectrum.
The zero slice of the motivic sphere 𝟙_X is constructed as a strict cofibrant ring spectrum ℍℤ_X^{slice}. The authors prove that ℍℤ_X^{slice} is stable under pull‑back (g⁎ℍℤ_{Spec k}^{slice} ≅ ℍℤ_X^{slice}) and that every slice sₙ(E) of any motivic spectrum E carries a canonical strict ℍℤ_X^{slice}‑module structure. This strict module structure is obtained at the model‑category level, guaranteeing homotopical rigidity needed for later constructions such as transfers.

Rational Coefficients and Transfers.
When X is geometrically unibranch and coefficients are rationalized, the paper invokes the work of Cisinski–Déglise on rational motivic categories. It shows that
 ℍℤ_X^{slice} ⊗ ℚ ≅ ℍℤ_X ⊗ ℚ,
i.e., the rational zero slice coincides with Voevodsky’s rational motivic cohomology spectrum. Moreover, the rational slices inherit transfer (norm) maps, establishing the “slice conjecture with transfers” proposed by Voevodsky. Consequently, all slices become objects in the triangulated category of rational motives equipped with the full six‑functor formalism.

Consequences and Conjectures Resolved.
The combination of pull‑back compatibility, explicit slice calculations for KH, and the identification of the rational zero slice settles several of Voevodsky’s conjectures:

1. The zero slice of the sphere spectrum is the motivic cohomology spectrum ℍℤ (integrally) and ℍℤ ⊗ ℚ (rationally).
2. All slices admit a canonical ℍℤ‑module structure and, after rationalization, admit transfers.
3. The slice filtration behaves well under arbitrary base‑change for schemes of finite type over a field with resolution of singularities.

Outlook.
The authors suggest that the pull‑back functoriality opens the door to relative motivic homotopy theory, allowing one to study slices over more general bases, to compare slices for different cohomology theories, and to explore further connections with algebraic cycles, mixed motives, and the emerging theory of motivic spectral algebra. They also note that extending these results to positive characteristic without resolution of singularities remains a challenging but promising direction.