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.

中文翻译：

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场上光滑仿射亨斯对的动机刚性

令$Z\to X$ 是域$k$ 上的平滑仿射方案的封闭浸没式，令$X^h_Z$ 表示$X$ 沿$Z$ 的汉塞尔化。我们证明 $E(X^h_Z)\simeq E(Z)$ 对于每个附加预层 $E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$ on $k$ 上的稳定动机同伦范畴是 $l_\varepsilon $-torsion 或 $l$-torsion，其中 $l\in \mathbb Z$ 与 $\operatorname {char} k$ 互质，并且 $l_\伐瑞西隆 =\sum _{i=1}^n \langle (-1)^i \rangle $。更一般地，同构适用于 $\textrm {Sm}_{k}$ 上的任何同伦不变 $l_\varepsilon $-torsion 或 $l$-torsion 线性 $\sigma $-quasi-stable 框架加性预层。这概括了先前已知的局部方案的结果。我们通过代数几何构造（稳定的）${\mathbb {A}}^1$-motivic 空间的同伦来证明上述同构。