I am sorry, if this a very standard fact.

Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\times_S S'$ is reduced?

If I need to assume something please suggest.(But in my case $f$ is not flat). On the other case, if you know something more general it also would be interesting.

  • 1
    $\begingroup$ No. Take $X=\{0\}\subset S=\mathbf A^1$ and $\psi:S'=\mathbf A^1\to S:t\mapsto t^2$. Probably $\psi$ should be smooth for this to work. $\endgroup$ Oct 20, 2023 at 13:32
  • $\begingroup$ Yes, let us assume $\psi$ to be smooth. $\endgroup$ Oct 20, 2023 at 13:35
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    $\begingroup$ If $\psi$ is smooth, then so is the projection $X \times_S S' \to X$, so the result follows from Tag 033B. In fact, in this case $X \times_S S'$ should be 'as smooth as $X$', e.g. it is normal if $X$ is, smooth if $X$ is, etcetera. $\endgroup$ Oct 20, 2023 at 13:51
  • $\begingroup$ Thank you! This completely answer my question. $\endgroup$ Oct 20, 2023 at 13:57


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