Resolvability of smooth schemes over non-perfect bases

Determine whether every smooth scheme over a non-perfect field or a Dedekind domain has a resolvable motive, where resolvable motives are generated from Tate twists of smooth projective schemes by finite limits, finite colimits, and retracts.

Background

The paper defines resolvable motives as the smallest subcategory of motivic modules closed under finite limits, finite colimits, and retracts that contains the perfect pure motives generated by smooth projective schemes. Over perfect fields, strong resolution of singularities implies that smooth varieties admit simple normal crossings compactifications and hence have resolvable motives.

The authors note that this implication is unclear over non-perfect fields and Dedekind domains because relative simple normal crossings compactifications are unavailable in that generality. They specifically point out that even smooth affine curves over imperfect fields may admit no smooth compactification, leaving the broader resolvability question unresolved.

References

However, it is unclear if one should expect all smooth schemes to have resolvable motives over non-perfect fields or over Dedekind domains. Relative sncd compactifications do not exist in this generality: already over an imperfect field, there are smooth affine curves which admit no smooth compactification at all Remark~4.14.

— Grothendieck ring of varieties and at-the-characteristic numerical invariants  (2610.02690 - Annala et al., 2 Oct 2026) in Remark in Section 2.1, immediately after Example 2.4

It is not clear if the map $K_0(Varres_S) \to K_0(Var_S)$ has any good surjectivity or injectivity properties. However, we make the following conjecture. If $k$ is a perfect field then all finite-type $k$-schemes are resolute. In particular, the map $K_0(Varres_k) \to K_0(Var_k)$ is an isomorphism.

— Grothendieck ring of varieties and at-the-characteristic numerical invariants  (2610.02690 - Annala et al., 2 Oct 2026) in Conjecture 2.12, Section 2.2, labeled \cref{conj:everything_is_resolute}