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Let $X$, $Y$ and $Z$ be Noetherian schemes.

If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times_X Z \to X$ is smooth/étale/unramified everywhere?

If not, which results can we obtain?

(In his textbook on Algebraic Geometry, Liu asks to prove that the answer is always "yes"...)

EDIT. So, indeed, the problem statement in the book is wrong...
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Let $X$, $Y$ and $Z$ be Noetherian schemes.

If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times_X Z \to X$ is smooth/étale/unramified everywhere?

If not, which results can we obtain?

(In his textbook on Algebraic Geometry, Liu asks to prove that the answer is always "yes"...)
show/hide this revision's text 2 edited body

Let $X$, $Y$ and $Z$ be Noetherian schemes.

If $f: Y \to X$ and $g: Z \to X$ are morphisms of finite type, such that at each point of $Y$, X$, at least one of the two morphisms is smooth/étale/unramified (at all points of its inverse image), can we conclude that the induced morphism $Y \times_X Z \to X$ is smooth/étale/unramified everywhere?

If not, which results can we obtain?
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