TITLE: Etale morphism and reduced schemes
QUESTION [5 upvotes]: Let $f:X \to Y$ be an etale morphism of Noetherian schemes. Is it true that the induced morphism on the reduced schemes, i.e., $f_{\mathrm{red}}:X_{\mathrm{red}} \to Y_{\mathrm{red}}$ is etale as well?

REPLY [9 votes]: Yes.
Firstly, recall that the base-change of an etale morphism is etale,
so  $X' := X \times_Y Y_{\text{red}} \to Y_{\text{red}}$ is etale.
Also, the base-change of a closed immersion is a closed immersion,
and so $X' \to X$ is a closed immersion.
Next, note that $X_{\text{red}} \to Y$ factors through each of $X$ and $Y_{\text{red}}$, and so factors through $X'$.  In other words, the closed immersion
$X_{\text{red}} \to X$ factors through $X'$.
Finally, recall that if $T \to S$ is etale and $S$ is reduced, then $T$ is reduced,
and so $X' \to X$ factors through $X_{\text{red}}$.    Thus $X' = X_{\text{red}}$,
and so $X_{\text{red}} \to Y_{\text{red}}$ is etale, as claimed.