It is well known that fibre product of noetherian schemes may not be noetherian. The inverse image of a coherent sheaf is coherent provided the domain is locally noetherian and codomain is noetherian. So my question has to do with the inverse image of coherent sheaves. Is it true that fibre product of noetherian schemes over a noetherian base is locally noetherian?
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$\begingroup$ It's not an exact duplicate, but it answers your question. $\endgroup$– jgonNov 7, 2018 at 16:28
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$\begingroup$ @jgon Thanks for your response. I don't see why the linked question answers that fibre product of noetherian schemes over a noetherian base is or is not locally noetherian. What am I missing here? $\endgroup$– user567863Nov 8, 2018 at 0:07
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1$\begingroup$ The answer to the linked question gives an example of a field extension $K/k$ such that $K\otimes_k K$ is not Noetherian. In the language of schemes, this is an example such that $\newcommand\Spec{\operatorname{Spec}}\Spec K\times_{\Spec k} \Spec K$ is not locally Noetherian. It should be clear that $\Spec F$ is Noetherian for any field $F$. I.e. it is an example of a fiber product of Noetherian schemes over a Noetherian base which is not locally Noetherian. $\endgroup$– jgonNov 8, 2018 at 0:11
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$\begingroup$ I see. Thank you very much ! $\endgroup$– user567863Nov 8, 2018 at 1:05
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