When is a morphism proper? - MathOverflow

When is a morphism proper?

2010-04-10T01:03:16Z

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.

Answer by Charles Staats for When is a morphism proper?

2010-04-10T02:53:02Z

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.

Answer by Barbara for When is a morphism proper?

2010-07-05T14:43:39Z

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.