Say $X \to Y$ is a surjective map of algebraic varieties, and $Z \subset Y$ is nonreduced. Then is the preimage $Z \times\_Y X$ also nonreduced?



Is the preimage of a nonreduced subscheme nonreduced?
In Allen's notation, take:


$R = k[t]$, $X = \operatorname{Spec} R$


$S = k[x,y]/(y^2 - x^2(x-1) )$, $Y= \operatorname{Spec} S$.


with the map defined by:


$y = t(t^2+1)$


$x=(t^2+1)$


$I=(x)$, $Z = \operatorname {Spec} S/I$.


$S/I$ contains a nilpotent, $y$ so $Z$ is non-reduced. $X \to Y$ is a surjective map of varieties. $X \times\_Y Z = \operatorname {Spec} R/RI = \operatorname{Spec} k[t]/(t^2+1)$, which is reduced.


