Let $X$ be a Noetherian scheme. Given two morphisms of schemes $\pi:T\to S$ and $f:X\to S$, if $\pi$ is flat, is $X\times_ST$ a Noetherian scheme?

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    $\begingroup$ I think math.stackexchange.com/questions/1154263/… should answer your question. In particular, there are examples where $S = \mathrm{Spec}(k)$, and $T = X = \mathrm{Spec}(K)$ for some field extension $K/k$. In this case, both $T$ and $X$ are Noetherian, and $\pi$ is (faithfully!) flat, so it's not even true with stronger assumptions, if I'm not mistaken. $\endgroup$ Aug 20, 2018 at 16:12


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