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Dear all,

Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$.

I would like to show that $f|_{U} : U \rightarrow V$ is again proper.

I have found a proof, by exploiting $U = X \times_Y V$ to show that $f_U$ is universally closed. It then follows. My question however is the following.

From Hartshorne II 4.8(f), we know that properness is local on the base, i.e. a moprphism is proper if and only if the target can be covered by opens $V_i$ such that $f^{-1}(V_i) \rightarrow V_i$ is again proper.

Question: Does that imply that this holds for any open $V \subseteq Y$, hence proving what i want in a way quicker way?

I am aware that that many properties of morphisms of schemes are defined using an open cover, and that under some circumstances this is equivalent to demanding it for all open covers. However i have never found the time to look into that properly, so apologies if this is very easy.

Thanks a lot!

flag	asked Dec 18 at 14:03 Joachim 115●5

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Dear Joachim, The base-change of a proper morphism is proper, and so in particular its restriction to $V \subset Y$ is proper. This is a natural way to argue, and I'm not sure why you object to it. If you really want an alternative, you could use the valuative criterion, for example. For a projective morphism, you could also argue as follows: factor $f: X \to Y$ as $X \hookrightarrow \mathbb P^n \times Y \to Y$ (the first map, which is a closed immersion, being the product of a projective embedding of $X$ and the map $f$, the second map being the projection). Now $U$ is equal to ... – Emerton Dec 18 at 15:34

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... $X \cap \mathbb P^n \times V$ (the intersection taking place in $\mathbb P^n \times Y$), and so $f_U$ factors as the composition of the closed immersion $V \hookrightarrow \mathbb P^n \times U$ and the projection $\mathbb P^n \times U \to U$. So you are reduced to checking this last map is proper, which you probably know. Regards, Matthew – Emerton Dec 18 at 15:35

Dear Matthew, i actually came only to post that i just found out about this simple application of what i already knew (the thing that you mentioned in the first sentence). I overlooked that possiblity. Thanks a lot for your answer anyway, since the valuative criterion application is helpful nonetheless. I might delete this question, i doubt if it will help anyone else.. – Joachim Dec 18 at 16:14

Quicker way to show that the restriction to a open subvariety is again proper?

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