Timeline for Differential map of a dominant morphism in char zero

6 events

when toggle format	what		by	license	comment
May 21, 2017 at 9:18	vote	accept	Heitor
May 20, 2017 at 9:41	vote	accept	Heitor
May 20, 2017 at 9:41
May 19, 2017 at 12:16	answer	added	Jason Starr		timeline score: 1
May 19, 2017 at 10:40	comment	added	Jason Starr		You can prove that by Noetherian induction. In the usual way, you reduce to the case that $X$ is integral. By generic smoothness, there exists a dense open subset $V$ of $X$ on which $\pi$ is smooth. If the closed complement $Z$ of $V$ does not dominate $Y$, then choose $U$ to be the open complement of the closure of $\pi(Z)$. If $Z$ does dominate $Y$, then by the induction hypothesis, there exists a dense open subset $U$ of $Y$ such that $Z\times_Y U \to U$ has surjective $d\pi$. Since also $d\pi$ is surjective on $V\times_Y U$, the result is proved by Noetherian induction.
May 19, 2017 at 9:58	comment	added	Francesco Polizzi		If $\pi$ is generically unramified and $Y$ is smooth, there is an exact sequence $$ 0 \longrightarrow T_X \stackrel{d \pi}{\longrightarrow}\pi^T_Y \to N_{\pi} \longrightarrow 0,$$ where $N_{\pi}=T_{X/Y}$ if $X$ is also smooth. See [Sernesi, Deformations of algebraic schemes*, p. 162].
May 19, 2017 at 9:43	history	asked	Heitor	CC BY-SA 3.0