Motivic Rigidity for Smooth Affine Henselian Pairs over a Field.

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. [ABSTRACT FROM AUTHOR]