关于平滑曲线的零稳定$\mathbb{A}^1$ -同伦群
On the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve
摘要：
给出了无限完满域上平滑曲线的第零稳定$\mathbb{A}^1$ -同伦群的上同调解释.我们证明了这个群同构于与第一Milnor-Witt $\mathrm{K}$ -理论层密切相关的某个层的第一Nisnevich（或Zariski）上同调群。这个上同调群可以用一个显式的Gersten型复形来计算。证明了如果基场代数闭，则平滑曲线的第零稳定$\mathbb{A}^1$ -同伦群与Suslin和Voevodsky用相对Picard群确定的第零Suslin同调群重合.作为结果，我们重新获得一个版本的Suslin的刚性定理。
英文摘要：
We provide a cohomological interpretation of the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor--Witt $\mathrm{K}$-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.
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