arXiv 2003.068922020-03-15math.AG

无限完满域上开光滑概形对的运动Segal定理

A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field

Aleksei Tsybyshev

备注：: 35 pages

摘要：

V. Voevodsyky奠定了去环动机空间的基础，以提供一个新的，更计算友好的，稳定的动机范畴$SH(k)$，G。加库沙和我。Panin使这个项目成为现实，同时与A。Ananievsky，A. Neshitov和A.德鲁日宁特别是G.加库沙和我。Panin证明了对一个无限完满域$k$和任何$k$ -光滑概型$X$，motivic空间$C_*Fr(X)\to Ω^{\infty}_{\mathbb{P}^1} Σ^{\infty}_{\mathbb{P}^1} (X_+)$的正则态射是Nisnevich-局部群完备的.在目前的工作中，该定理的推广的情况下，光滑开放对$(X,U),$其中$X$是一个$k$ -光滑计划，$U$是它的开放子计划相交的每个组成部分的$X$在一个非空子计划。我们证明了在这种情况下，动机空间$C_*Fr((X,U))$是Nisnevich-局部连通的，动机空间态射$C_*Fr((X,U))\to Ω^{\infty}_{\mathbb{P}^1} Σ^{\infty}_{\mathbb{P}^1} (X/U)$是Nisnevich-局部弱等价的.此外，我们还证明了如果$X$的每个分支的$S=X-U$的余维数大于$r \geq 0,$，则单纯层$C_*Fr((X,U))$是局部$r$连通的.

英文摘要：

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.

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英文摘要：

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