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In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:

Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets represents the functor

$$F_X^m: (\operatorname{Sch}/k) \to (\operatorname{Set}), Z \mapsto \operatorname{Hom}(Z \times \operatorname{Spec} \ k[t]/(t^{m+1}), X)$$

i.e. for every $Z$ we have $F^m_X(Z) \cong \operatorname{Hom}(Z,X_m)$.

By construction the canonical quotient map $k[t]/(t^{m+1}) \to k[t]/(t^{m})$ induces natural morphism between functors $F_X^m \to F_X^{m-1}$. By Yoneda this induces a unique morphism $\psi_{m, m-1}: X_m \to X_{m-1}$.

Question: Assume that $X$ is non singular (=smooth). Why this implies that $\psi_{m, m-1}$ is a smooth map? (compare Example 1.6 on p 3)

In case $X= \mathbb{A}^n$ one can show that $X_m= \mathbb{A}^{(m+1)n}$ and that $\psi_{m, m-1}: \mathbb{A}^{(m+1)n} \to \mathbb{A}^{mn}$ is the canonical projection. It is smooth.

What about the general case if $X$ smooth. How to prove the smothmess of $\psi_{m, m-1}$? Can the claim be reduced to the case above with affine space?

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    $\begingroup$ 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. $\endgroup$
    – Samir Canning
    Commented Mar 28, 2020 at 22:54
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    $\begingroup$ 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. $\endgroup$
    – user267839
    Commented Mar 28, 2020 at 23:16

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