Motivic Rigidity for Smooth Affine Henselian Pairs over a Field

Abstract

Abstract Let $Z\to X$ be a closed immersion of smooth affine schemes over a field $k$, and let $X^h_Z$ denote the henselisation of $X$ along $Z$. We prove that $E(X^h_Z)\simeq E(Z)$ for every additive presheaf $E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$ on the stable motivic homotopy category over $k$ that is $l_\varepsilon $-torsion or $l$-torsion, where $l\in \mathbb Z$ is coprime to $\operatorname {char} k$, and $l_\varepsilon =\sum _{i=1}^n \langle (-1)^i \rangle $. More generally, the isomorphism holds for any homotopy invariant $l_\varepsilon $-torsion or $l$-torsion linear $\sigma $-quasi-stable framed additive presheaf on $\textrm {Sm}_{k}$. This generalises the result known earlier for local schemes. We prove the above isomorphism by constructing (stable) ${\mathbb {A}}^1$-homotopies of motivic spaces via algebraic geometry. To achieve this, we replace Quillen’s trick with an alternative and more general construction that provides relative curves required in our setting.