Generically finite morphisms

Certain results have a variant for generic points, and a variant which works over a dense open. As an example let’s discuss “generically finite morphisms” of schemes.

The first variant is Lemma Tag 02NW: If f : X —> Y is of finite type and quasi-separated, η is a generic point of an irreducible component of Y with f^{-1}(η) finite, then there exists an affine open V of Y containing η such that f^{-1}(V) —> V is finite.

The second variant is Lemma Tag 03I1: If f : X —> Y is a quasi-finite morphism, then there exists a dense open V of Y such that f^{-1}(V) —> V is finite.

Comments: (a) In the second variant it isn’t necessarily the case that every generic point of every irreducible component of Y is contained in the open V, although this would follow from the first variant if we assumed f quasi-separated. (b) The proof of the first variant in the stacks project is basically elementary; the proof of the second variant currently uses (a technical version of) Zariski’s main theorem.

The point I am trying to make (badly) is that you can often get around making any separation assumptions by trying to prove a variant “over a dense open”. Maybe the archetype is the following result (Lemma Tag 03J1): Every quasi-compact scheme has a dense open subscheme which is separated.

9 thoughts on “Generically finite morphisms

  1. Johan,

    Here is a stacks question that came up at a workshop last week. Let X be a smooth, quasi-compact Artin stack over an algebraically closed field k. Does there exist a global quotient stack Y and a 1-morphism f:Y–>X which is representable by surjective, ‘etale morphisms? Do we even know this for the Artin stack of prestable curves (with a bounded number of irreducible components)?

  2. Johan, here’s a dumb question. What is the *definition* of quasifinite morphism? I could imagine several possible choices. In the stacks project, you have a section on generically finite morphism, but you (likely intentionally) don’t give a definition. Or do you mean something in that section (e.g. (1)=(2) in tag02NW) should be the definition? You can answer by email if you want; I may forget to look at this page again soon.

    • A morphism of schemes is quasi-finite if it is of finite type and every point is isolated in its fibre, see Definition Tag 01TD. The last condition is equivalent with requiring the fibres to be finite (see Lemma Tag 02NH, which is your choice in the FOAG lecture notes I think.

    • Sorry, I was being a moron — I meant to ask what a *generically finite* morphism is.

      • Aha, now the question makes much more sense! Yes, not having the definition was intentional, because I don’t know what the right definition is in very general situation. I’m already conflicted over whether the property should be local on the source or on the target. Thus I decided to just have a bunch of lemmas around the concept.

        Do you have a suggestion?

        • I don’t, which is actually why I don’t have any definition currently in the “Foundations of Algebraic Geometry” notes — the same reason as you. I’m thus disturbed that this notion gets stated in talks when it is not clear to me that the audience has any agreement on what it means. One possibility: to state that in the special case of f: X \rightarrow Y where Y is irreducible, and f is locally of finite type, the requirement is that the generic point have finitely many preimages (part of tag 02NW), and this has an obvious extension when Y is, for example locally Noetherian (so you just look at all the generic points of irreducible components), but that there are different potentially useful extensions to more general situations (followed by a link to the results in that section of the stacks project).

          • OK, that is fine, and if I read it right, then you are choosing it to be a local on the base. But this means that the map P^1 —> A^1 which maps P^1 to 0 ∈ A^1 would be generically finite. And somehow that doesn’t feel right!

          • I had always thought of this as generically finite. I can see why you find it weird. I find it weird, but it doesn’t bother me too much. Most of the fibers are indeed finite, and sometimes infinite. I know that isn’t convincing though!

          • OK, no problem. I think your definition is fine. But… the following alternative just occurred to me: “there exists a dense open U of X such that U —> Y is locally quasi-finite”. This is local on the source, and it implies your condition if the morphism is quasi-compact. What is weird about this alternative is that it doesn’t clearly relate to the morphism being finite (over something).

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