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Suppose $X$ and $Y$ are regular schemes (i.e. all local rings are regular) and flat over some base $S$. Assume further that $Y$ has relative dimension $0$ over $S$. Does it follow that $X \times_S Y$ is regular?

I'm really interested only in the special case that $S = Spec \ \mathbb Z$ and $Y = Spec (\mathcal O_L)$ where $\mathcal O_L$ is the ring of integers of a number field, so any suggestions that speak to this case would be most welcome!

regular
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    Look at the example $L=\mathbb Q[\sqrt{2}]$ and $X=Y$. –  Jul 23 '13 at 07:59
  • Thanks, this is such a simple counterexample that I feel guilty I didn't try it. I guess the intuition is that Y is not smooth at 2? – regular Jul 23 '13 at 08:37
  • Yes. In general, if you want $X\times_S Y$ be regular, you almost need $X, Y, S$ be regular and $X$ or $Y$ be smooth over $S$. –  Jul 23 '13 at 08:55
  • How about the following amendment? Suppose that for every point s in S, there is a neighbourhood U in S such that at least one of X_U or Y_U is smooth over U. Then I would guess we could conlude the fibre product X x Y is regular (since smooth base change preserves regularity). Would this work? Again, thanks so much for the help. – regular Jul 23 '13 at 09:03
  • Yes, with the hypothesis on $X, Y, S$ in my previous comment. This is because regularity is a local condition. –  Jul 23 '13 at 09:05
  • May I ask you something? How can I show that $\mathbb{Z}[\sqrt{2}] \otimes \mathbb{Z}[\sqrt{2}]$ is not regular ring? I don't know what is prime ideal of $\mathbb{Z}[\sqrt{2}] \otimes \mathbb{Z}[\sqrt{2}]$ and how to localize at this prime ideal (to show regular local ring). – hew Mar 01 '21 at 14:00

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