2018/09/11 by A. Druzhinin, Druzhinin, A.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1809.04158
openalex publication_date 2018/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Z→ X be a closed immersion of smooth affine schemes over an arbitrary field k, and XhZ denote the henselization of X along Z. For each presheaf E\colon SH(k)→ Abop on the stable motivic homotopy category over k and the induced continuous presheaf E\colon EssSmk→ Abop on the category of essentially smooth schemes there is a homomorphism E(XhZ)→ E(Z). We prove that this is an isomorphism for any lε-torsion presheaf E, for l∈ \mathbb Z, (l,chark k)=1, and lε=∑i=1n ⟨ (-1)i ⟩. More generally, the isomorphism holds for any homotopy invariant lε-torsion linear σ-stable framed additive presheaf F over k. The case of l-torsion presheaves follows as well. The result generalises known Gabber's rigidity theorems for local henselian schemes to the case of smooth affine henselian pairs. The above isomorphism is proven by constructing of (stable) \mathbbA1-homotopies of motivic spaces via algebro-geometric techniques. To achieve this in our setting we replace often used Quillen's trick by an alternative construction that provides required smooth relative curves over smooth affine schemes for an arbitrary base field.