Rigidity for smooth affine pairs over a field

Abstract: Let $Z\to X$ be a closed immersion of smooth affine schemes over an arbitrary field $k$, and $X^h_Z$ denote the henselization of $X$ along $Z$. For each presheaf $E\colon \mathbf{SH}(k)\to \mathrm{Ab}^\mathrm{op}$ on the stable motivic homotopy category over $k$ and the induced continuous presheaf $E\colon \mathrm{EssSm}k\to \mathrm{Ab}^\mathrm{op}$ on the category of essentially smooth schemes there is a homomorphism [E(X^h_Z)\to E(Z).] We prove that this is an isomorphism for any $l\varepsilon$-torsion presheaf $E$, for $l\in \mathbb Z$, $(l,\mathrm{chark}\,k)=1$, and $l_\varepsilon=\sum_{i=1}^n \langle (-1)^i \rangle$. More generally, the isomorphism holds for any homotopy invariant $l_\varepsilon$-torsion linear $\sigma$-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) $\mathbb{A}^1$-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.