标题： 概型的同伦理论与$R$-等价性 显示英文标题

标题： Homotopy theory of schemes and $R$-equivalence

摘要： 我们证明了，对于域 $X$ 上任意光滑射影概形 $k$，当特征为 char. 时，有 $0$，Spec $k$到 $X$在 $\mathbf{A}^1$-同伦范畴中的态射集，在 $\mathcal{H}_{\mathbf{A}^1}(k)$的概形同伦范畴中与 $X(k)$对 $R$-等价关系的商集一一对应，并且是 $X$的双有理不变量。 这是通过建立平滑的$k$-概型范畴通过双有理映射局部化与范畴$\mathcal{H}_{\mathbf{A}^1}(k)$之间的精确关系，并应用第二作者和 R. Sujatha 关于双有理不变量的结果来实现的。这为 A. Asok 和 F. Morel 所得结果给出了一个新的证明。 显示英文摘要

摘要： We prove that, for any smooth and projective scheme $X$ over a field $k$ of char. $0$, the set of maps from Spec $k$ to $X$ in the $\mathbf{A}^1$-homotopy category of schemes $\mathcal{H}_{\mathbf{A}^1}(k)$ is in bijection with the quotient of $X(k)$ by $R$-equivalence, and is a birational invariant of $X$. This is achieved by establishing a precise relation between the localization of the category of smooth $k$-schemes by birational maps and the category $\mathcal{H}_{\mathbf{A}^1}(k)$, and by applying results of the second named author and R. Sujatha on birational invariants. This gives a new proof of results obtained by A. Asok and F. Morel.