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A morphism of varieties over $\mathbb{C}$, $f:V\to W$ is proper if it is universally closed and separated. One way to check properness is the valuative criterion.

What other methods do we have for determining if a morphism is proper? Particularly, I'm interested in quasi-projective varieties, but ones that aren't actually projective. And while a completely algebraic, valid over all fields or for schemes answer would also be good, I'm looking at complex varieties, and may be able to assume that the singularities are all finite quotient singularities.

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if direct image functor of a morphism preserves coherent sheaves on noetherian schemes, then this morphism is proper. The other direction is well known –  Shizhuo Zhang Apr 10 '10 at 1:46
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It's equivalent for the analytified map to be proper in the usual topological sense; see the Expose on analytic/algebraic stuff near the end of SGA1. This allows arbitrary schemes of finite type. –  BCnrd Apr 10 '10 at 2:52
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2 Answers

up vote 7 down vote accepted

Assume $V$ and $W$ are quasiprojective. Let $i:V\to X$ be a locally closed embedding with $X$ projective (for instance $X$ could be $P^n$). Consider the induced map $g:V\to X\times W$; this is also a locally closed embedding. Then $f$ is proper iff $g$ is a closed embedding, or equivalently if $g(V)$ is closed.

As for the topological approach, use the definition of properness given by Charles Staats. Let $f:X\to Y$ be a continuous map of Hausdorff second countable topological spaces. The base change $f':X'\to Y'$ of $f$ by a continuous map $g:Y'\to Y$ is defined by letting $X'$ be the set of pairs $(x,y')$ in $X\times Y'$ such that $f(x)=g(y')$ (with the induced topology from $X\times Y'$), and $f':X'\to Y'$ the obvious projection. Then $f$ is proper if and only if all its base changes are closed. This may not be logically relevant, but I find it very comforting. To connect the two cases note that, given a locally closed embedding of complex algebraic varieties, it is closed in the Zariski topology iff it is closed in the Euclidean topology.

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There's a purely topological notion of properness, in which a continuous map is proper if and only if the inverse image of every compact set is compact. I have been told that this corresponds with the algebraic notion in the case of complex varieties, although I do not have a reference.

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I guess one can find the reference in Serre's GAGA, If I remember correctly, he attributed this result to Chow. –  Li Yutong May 8 '13 at 19:40
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