论文标题
通用的Birational不变性和$ \ Mathbb {a}^1 $ - 本体
Universal birational invariants and $\mathbb{A}^1$-homology
论文作者
论文摘要
让$ k $成为承认奇异性解决方案的领域。在本文中，我们证明了Zeroth $ \ Mathbb {a}^1 $ - 人体学$ \ Mathbf {h}^{\ Mathbb {a}^1} _0 $作为平滑适当$ k $ -varieties的函数公平不变的函数公平的函数公平，是abelian by abelian by abeLian for abeLian for的价值。对于平稳的$ k $ -varietio $ x $，我们还证明了$ \ mathbf {h}^{\ mathbb {\ mathbb {a}^1} _0（x; \ mathbb {q}）（\ mathrm {spec} k）$ come of $ r $ - equealceRenceSceRce $ x的维度。我们将这些结果推论为结构定理的后果，即对于平滑的$ k $ - variety $ x $，捆绑$ \ mathbf {h}^{\ mathbb {a}^1} _0} _0（x）$是免费的Abelian Presheaf。 $π_0^{b \ Mathbb {a}^1}（x）$ asok-morel。
Let $k$ be a field admitting a resolution of singularities. In this paper, we prove that the functor of zeroth $\mathbb{A}^1$-homology $\mathbf{H}^{\mathbb{A}^1}_0$ is universal as a functorial birational invariant of smooth proper $k$-varieties taking values in a category enriched by abelian groups. For a smooth proper $k$-variety $X$, we also prove that the dimension of $\mathbf{H}^{\mathbb{A}^1}_0(X;\mathbb{Q})(\mathrm{Spec} k)$ coincides with the number of $R$-equivalence classes of $X(k)$. We deduce these results as consequences of the structure theorem that for a smooth proper $k$-variety $X$, the sheaf $\mathbf{H}^{\mathbb{A}^1}_0(X)$ is the free abelian presheaf generated by the birational $\mathbb{A}^1$-connected components $π_0^{b\mathbb{A}^1}(X)$ of Asok-Morel.
