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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...

3 added 126 characters in body; deleted 31 characters in 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 $X$, 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"...)

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?

1

# products and smooth/étale/unramified morphisms

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 $Y$, 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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