论文标题
天真
Naive $\mathbb A^1$-homotopies on ruled surfaces
论文作者
论文摘要
我们明确地描述了从光滑的Henselian本地方案到平滑的投影表面，$ \ Mathbb A^1 $ - 链同质型形态类别，该表面在属于$> 0 $的曲线上进行了birationally的统治。因此，我们确定了这种表面的幼稚$ \ mathbb a^1 $连接的组件的捆，并表明当表面不是最小模型时，它与其真正的$ \ mathbb a^1 $连接的组件不一致。但是，在尺寸$ \ leq 1 $同意的方案上，幼稚和真实$ \ mathbb a^1 $连接的组件的束带的截面既有。结果，我们表明，在光滑的投射表面上的莫雷 - 伏娃娃（Morel-Voevodsky）奇异结构，该结构在$> 0 $的曲线上被统治，如果表面不是最小的模型，则不是$ \ mathbb a^1 $ local。
We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb A^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb A^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb A^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, we show that the Morel-Voevodsky singular construction on a smooth projective surface, which is birationally ruled over a curve of genus $> 0$, is not $\mathbb A^1$-local if the surface is not a minimal model.
