AI 中文总结
该研究将 n-双有理动机同伦范畴提升为预层，推导单分支概型双有理动机同伦范畴的分解性质，给出双有理等价的检测方法，并证明稳定双有理态射等诱导完全忠实嵌入。
AI 中文摘要
我们将 n-双有理动机同伦范畴 $S\mapsto \mathcal{H}^n(S)$ 提升为 $Corr(\mathrm{Sch})_{uglt,sm}$ 上取值于 $Pr^L$ 的预层。由此，对任何单分支概型 $X$，其（零次）双有理动机同伦范畴 $\mathcal{H}^{b\mathbb{A}^1}(X)$ 可分解为其函数域的双有理动机同伦范畴的乘积；特别地，对簇 $V$，有 $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$。这意味着 $Sm_S$ 中单分支概型的双有理等价可通过其一般纤维的双有理可缩性检测。最后，我们证明稳定双有理态射与纯超越域扩张诱导双有理动机同伦范畴的完全忠实嵌入。
英文摘要
We promote the $n$-birational motivic homotopy category assignment $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, the birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of schemes in $Sm_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.
