Algebraic Geometry – Why Unramified Maps Are Not Locally of Finite Presentation

ag.algebraic-geometry

I have read and heard several times that it is “important” that unramified maps are not required to be locally of finite presentation, but only locally of finite type.

Apart from this issue with unramified I have always preferred “locally of finite presentation” as it seems to be conceptually and morally better then “locally of finite type”. It also has better categorical properties. In the Noetherian case they coincide, but outside the Noetherian world it seems to me that “locally of finite presentation” is the natural notion, and “locally of finite type” is a definition that just lacks a condition.

However, for some reason unramified maps need to be an exception.

EGA does define unramified maps as maps that are formally unramified and locally of finite presentation (as is the case for étale and smooth). But Raynaud diverges from this definition in his book “Anneaux locaux henséliens”. He requires only locally of finite type.

Johan de Jong writes on his Stacks Project blog {1}:

In only one case sofar have I changed the definition: namely David Rydh convinced me that we should change unramified to the notion used in Raynaud’s book on henselian rings (i.e., only require locally of finite type and not require locally of finite presentation).

Question: What is the justification for this definition? Is there some conceptual explanation of why this is the “right” definition? Are there important theorems that would be false otherwise, or become miserable to state or prove?

{1}: http://math.columbia.edu/~dejong/wordpress/?p=1049

Best Answer

Various people have imposed finite presentation instead of finite type: most notably Grothendieck for unramified and Demazure–Gabriel for proper maps. Both these cases seem to have been motivated by the simple functorial characterization of locally of finite presentation. I would however argue that finite presentation, as opposed to finite type, is not the most natural choice for immersions and unramified, proper, finite, quasi-finite, projective and quasi-projective morphisms. Let me try to motivate this:

The difference between finite type and finite presentation arises when you take the image of a map between finitely presented modules/algebras. For this reason it would be unnatural to demand that closed immersions (surjections) are of finite presentation, for example, the schematic image of a morphism $X\to Y$ between finitely presented schemes is a closed immersion $Z\hookrightarrow Y$ which need not be finitely presented. Since it is natural to include closed immersions among finite and proper morphisms, this motivates including maps of finite type.

Other constructions involving finite and proper morphisms also result in morphisms of finite type (also for quasi-finite morphisms etc):

  1. If you blow-up a scheme in a finitely generated ideal, you obtain a proper morphisms of finite type but it is not of finite presentation even if the ideal is finitely presented.
  2. Unions of finite extensions of rings are finite (but finite presentation need not be preserved).

Another natural source for morphisms (locally) of finite type that are not (locally) of finite presentation is the diagonal of any morphism.

A perhaps more compelling reason is that all the important results hold for finite type morphisms, including:

  1. Projective morphisms are universally closed.
  2. Quasi-finite morphisms factor as an open immersion followed by a finite morphism.
  3. Proper + quasi-finite = finite.

When it comes to unramified morphisms, one has several nice results that hold also for finite type morphisms:

  1. Description of monomorphisms, locally of finite type, in terms of fibers.
  2. Description of unramified morphisms, in terms of fibers.
  3. Description of unramified morphisms, in terms of vanishing of Kähler differentials.
  4. Description of unramified morphisms as having an étale diagonal (for maps between schemes: an open immersion).
  5. Description of unramified morphisms as being étale-locally a closed immersion.
  6. Description of unramified morphisms as factorizing canonically as a closed immersion followed by an étale morphism.

All these results fail if we remove "finite type" though. For étale morphisms, however, it is necessary to assume finite presentation and not merely finite type to get a well-behaved notion.

A morphism of finite type $f$ (over a quasi-compact and quasi-separated scheme) factors as a closed immersion $i$ followed by a finitely presented morphism $g$. Moreover, if $f$ is finite, proper, quasi-finite, etc., we may assume that $g$ is so as well. The difference between finite type and finite presentation is thus captured by closed immersions.

Let me also mention that neither finite type nor finite presentation is as well-behaved as finite type in the noetherian case. For example, neither finitely generated, nor finitely presented modules give abelian categories. Coherent modules and pseudo-coherent complexes are other variations that have their drawbacks and benefits. For example, in Grothendieck duality, neither proper nor proper and finite presentation is good enough. There are also situations were finite presentation and arbitrary nil-immersions are adequate (what I have called "constructible finite type").

Finally, there are also some situations where weaker notions than finite type are enough. In étale cohomology, proper base change holds after weakening finite type (any universal homeomorphism induces an equivalence of étale topoi). Chevalley's upper semicontinuity of fiber dimension holds for any universally closed, separated and quasi-compact morphism.

Related Question