论文标题
在无限的完美领域上的开放式平滑方案的动机定理
A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field
论文作者
论文摘要
V. Voevodsyky为动机空间奠定了基础，以提供一种新的，更适合计算的友好型，构建稳定的动机类别$ sh（k）$，G。Garkusha和I. Panin I. Panin，使该项目成为现实，同时与A. Ananievsky，A。A. Neshitov和A. Druzhinin合作。特别是，G。Garkusha和I. Panin证明，对于无限的完美字段$ k $和任何$ k $ -smooth方案$ x $ x $动机空间的规范形态σ^{\ infty} _ {\ mathbb {p}^1}（x _+）$是nisnevich，nisnevich-locally thl群组完成。 在目前的工作中，该定理对平滑的开放对$（x，u），$ x $是$ k $ -smooth方案的概括，$ u $是其开放式亚cheme，将$ x $的每个组件相交，而不是x $。我们声称，在这种情况下，动机空间$ c_*fr（（x，u））$是nisnevich-loccly连接的，动机空间形态$ c_*fr（（x，u））\ toω^{\ inpty} _ {\ infty} _ { （x/u）$是Nisnevich，在尼斯尼维奇大处是一个弱的等效性。此外，我们表明，如果$ x $的每个组件中的$ s = x-u $的编码大于$ r \ geq 0，则$ simplicial shipicial sheaf $ c_*fr（（x，u））$是本地$ r $连接的。
V. Voevodsyky laid the groundwork of delooping motivic spaces in order to provide a new, more computation-friendly, construction of the stable motivic category $SH(k)$, G. Garkusha and I. Panin made that project a reality, while collaborating with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field $k$ and any $k$-smooth scheme $X$ the canonical morphism of motivic spaces $C_*Fr(X)\to Ω^{\infty}_{\mathbb{P}^1} Σ^{\infty}_{\mathbb{P}^1} (X_+)$ is Nisnevich-locally a group-completion. In the present work, a generalisation of that theorem to the case of smooth open pairs $(X,U),$ where $X$ is a $k$-smooth scheme, $U$ is its open subscheme intersecting each component of $X$ in a nonempty subscheme. We claim that in this case the motivic space $C_*Fr((X,U))$ is Nisnevich-locally connected, and the motivic space morphism $C_*Fr((X,U))\to Ω^{\infty}_{\mathbb{P}^1} Σ^{\infty}_{\mathbb{P}^1} (X/U)$ is Nisnevich-locally a weak equivalence. Moreover, we show that if the codimension of $S=X-U$ in each component of $X$ is greater than $r \geq 0,$ the simplicial sheaf $C_*Fr((X,U))$ is locally $r$-connected.
