I have a question which is probably well known. Let $f:X\rightarrow Y$ be a birational morphism of reduced schemes and $Y_1\rightarrow Y$ be an étale morphism. Is it true that $X\times_Y Y_1\rightarrow Y_1$ is birational?
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What definition of birational are you using and do you have any other restrictions on your schemes or just reduced over $\mathbb{Z}$? – Dori Bejleri Dec 16 '13 at 20:11
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Dear @Dori: "reduced" is an absolute concept for a scheme $X$ , not depending on $X$ being over $\mathbb Z$ or over any other scheme. – Georges Elencwajg Dec 16 '13 at 20:44
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Oh I worded that wrong! I meant is he assuming anything else other than reduced for example is he just working over $\mathbb{Z}$ or over a field. The reason I asked is because I haven't seen people talk about birational morphisms other than for schemes over fields so I wasn't sure if the concept was still similar over more general bases. – Dori Bejleri Dec 16 '13 at 20:50
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@hjia: this is true for any open morphism $Y_1\to Y$, and étale morphisms are open. – Cantlog Dec 17 '13 at 21:05