Pullback of a reduced closed subscheme along an étale morphism Let $X$ any $Y$ be reduced schemes of finite type over a field and $W\subseteq Y$ a closed reduced subscheme.


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*What is an example of such $X$ and $Y$ and $W$ and of a morphism $f:X\to Y$ such that the closed subscheme $X\times_Y W\subseteq X$ is not reduced?

*Is there such an example for $f:X\to Y$ étale?


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*This even fails when $W$ is the spectrum of a field. Take any ramified morphism (such as $\mathbb{A}^1 \to \mathbb{A}^1, t \mapsto t^2$, the fiber at $0$ is the spectrum of $k[t]/(t^2)$).

*I don't think so. Since $X \times_Y W \to W$ is étale, it suffices to remark that for an étale morphism of local noetherian schemes $X \to Y$: If $Y$ is reduced, then also $X$ is reduced. This follows from Proposition 19.5 in Stacks project, etale morphisms.
