Timeline for Morphism between jet spaces smooth
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| Mar 28, 2020 at 23:16 | comment | added | user267839 | Yes, that's indeed a local problem, so locally we have an etale map $U \to \mathbb{A}^N$ for open affine $U \subset X$. And $Y_m\cong X_m\times_X Y$ follows from description of functor $F_Y^n(-)$. That is if $T= \operatorname{Spec} \ R$ is an affine test scheme we only need to lift a map $\operatorname{Spec} R[t]/(t^{n+1}) \to X$ to a map $\operatorname{Spec} R[t]/(t^{n+1}) \to Y$ but that's the etaleness! And the claim follows as base change preserve smoothness. I see, thank you. | |
| Mar 28, 2020 at 22:54 | comment | added | Samir Canning | It will follow from the affine case. You should show that if you have an étale morphism $Y\rightarrow X$, then $Y_m\cong X_m\times_X Y$. Use the smoothness hypothesis to come up with an appropriate étale morphism from a neighborhood of every point in $X$ to affine space. | |
| Mar 28, 2020 at 22:39 | history | asked | user267839 | CC BY-SA 4.0 |