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

  1. 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?
  2. Is there such an example for $f:X\to Y$ étale?
  1. 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 $k[t]/(t^2)$).

  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.

  • $\begingroup$ Dear @Martin Brandenburg, yes, thatnk you for answering the first point, this is a nice and easy example. Do you have an argument for 2.? $\endgroup$ – Ronald Bernard Apr 23 '13 at 22:42

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