is_dominant for morphisms of (some) schemes

[thofma]

In an application, I have a map $f \colon \mathbf{C}^n \to \mathbf{C}^m$ given by rational polynomials and I want to know if the image of $f$ is dense in $\mathbf{C}^m$. At the moment I have to do this by constructing morphisms of polynomial rings by hand and computing kernels. Thanks to @simonbrandhorst I now know that my original question is equivalent to the corresponding morphism of schemes being dominant. We can construct the affine schemes and the morphism. It would be great if we could also provide is_dominant for (some) morphism of schemes, including affine ones.

The stacks page on this topic might be useful.

CC: @simonbrandhorst

[afkafkafk13]

This is definitely an application which many algebraic geometers find useful. It is often nasty to do by hand for morphisms of covered schemes (with a slightly non-standard covering)

[HechtiDerLachs]

This also came up recently in work with @simonbrandhorst . I like the idea and, at least for varieties, this should be rather easy to implement. I guess that would already cover most use cases?

[wdecker]

The point here is that we need a consistent concept of rational maps and morphisms for schemes. At the moment, for example, the definition of morphisms for projective schemes does not make sense. Based on a consistent concept, we can then offer the usual things such as is_dominant or is_birational. I had some discussions with @jankoboehm and @HechtiDerLachs on this topic, but we finally need to sit down and work on it.