关于$\mathbb A^1$ -链连通分支构造迭代的注记
Remarks on iterations of the $\mathbb A^1$-chain connected components construction
摘要：
我们表明，层的$\mathbb A^1$ -连接组件的Nisnevich层的集合和它的通用$\mathbb A^1$不变商（通过迭代的$\mathbb A^1$ -链连接组件的建设和采取直接限制）同意域值点。这建立了任何空间的$\mathbb A^1$ -连通分支层的场值点的显式公式。给定任意自然数$n$，我们构造了一个$\mathbb A^1$ -连通空间，在该空间上朴素$\mathbb A^1$ -连通分支构造的迭代在$n$第阶段之前不稳定.
英文摘要：
We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $\mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $\mathbb A^1$-connected space on which the iterations of the naive $\mathbb A^1$-connected components construction do not stabilize before the $n$th stage.
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