# What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?

(I asked it first in MathStackExchange but I haven't get an answer yet)

Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms. For unramified morphisms, it is originally defined in EGA as locally finitely presented + formally unramified morphisms, but now they are widely accepted as locally of finite type + formally unramified morphisms.

My question is, why do we need to add the "locally finitely presented" or "locally of finite type" conditions in the "true definition" of smooth/étale/unramified morphisms?

According to vakil's discussion and this note about motivations of unramified morphisms, we can see that the three morphisms are analogues of some important notions in differential geometry:

1. Smooth-Submersions: surjections on tangent space, e.g. $$\mathbb{A}^9\to \mathbb{A}^5$$
2. Étale-Covering Spaces: bijections on tangent space, e.g. $$\mathbb{A}^5\to \mathbb{A}^5$$
3. Unramified-Immersions: injections on tangent space, e.g. $$\mathbb{A}^2\to \mathbb{A}^5$$

From my point of view, given a morphism of schemes $$f:X\to Y$$, the natural analogue of surjection (resp. bijection, resp. injection) on tangent spaces is perfectly described by surjection (resp. bijection, resp. injection) of $$\DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\Hom}{Hom}\Hom_Y(\Spec A,X)\to \Hom_Y(\Spec A/I,X)$$ where $$\Spec A$$ is any afine $$Y$$-scheme with $$I^2=0$$.

In the language of this note about motivations of unramified morphisms, they are all the "differential like data", and tangent vectors can be thought as differentals. So I would be happy to accept the above definitions as the defitions of smooth (resp. étale, unramified) morphisms.

Is there any natural motivations that we include these finiteness conditions? The idea "we need the fibres of smooth morphisms to be smooth varieties" is not enough to convince me, because there are still the case étale morphisms and unramified morphisms, also why do we need that naturally?

e.g.

• Is there any morphisms of schemes that are not expected to be smooth/étale/unramified intuitively but they fall into the cateogy of formally smooth/étale/unramified? So to exclude them we need to introduce finiteness condition.
• Is there any big theorems that have to include finiteness conditions?
• Maybe the true analogue indeed contains finiteness conditions from the begining?
• Chevalley's theorem guarantees that smooth and étale morphisms will be open maps (as they are flat, locally of fin pres), which isn't true for formally smooth/étale. – Harry Gindi Oct 3 '20 at 21:50
• @HarryGindi Thanks, I didn't get it the first time, then I googled that submersions and covering spaces are open maps, now I get it. – Z Wu Oct 5 '20 at 16:06
• I guess I'll add it as an answer since people agreed with it. – Harry Gindi Oct 5 '20 at 16:12
• It is often useful in geometry to be allowed to work with coordinates, at least locally. If you think of smooth morphisms as being (Zariski) locally defined by a system of equations satisfying the jacobian criterion, then you end up with the notion of smooth morphism with the classical finitess properties. Put in other words, having a perfect cotangent complex is a very useful property, and if you are smooth-like, such a cotangent complex will be concentrated in degree zero, hence a locally free module of finite type. Such properties are useful in many ways in practice. – Denis-Charles Cisinski Oct 5 '20 at 23:21
• @Denis-CharlesCisinski From Tag06B5, we know the cotanget sheaf of formally smooth morphism is locally projective, hence locally free. I believe this is nice enough to be called smooth-like. But the coordinate idea is good. – Z Wu Oct 6 '20 at 16:45

The main difference between formally smooth/smooth and formally étale/étale is that the formal versions don't need to be open maps. Chevalley's theorem on constructible topology says that a locally finite presentation map is an open map for the constructible topology, and if in addition you include flatness, you can show that it is an open map for the specialization topology. A map is open for the Zariski topology if and only if it is constructibly open and specialization-open.

As a consequence, smooth/étale maps (being flat and locally of finite presentation) are (universally) open maps.

There are other reasons involving Noetherian approximation to include finite presentation as well. Formally smooth/étale maps don't descend along inverse limits, while smooth and étale maps do. (In the qcqs case, that is).

• How about formally unramified/unramified morphisms? – Z Wu Oct 6 '20 at 16:35
• @LanceWu I think the answer by David Rydh to this question should be helpful for unramified maps: mathoverflow.net/questions/204437/… – Harry Gindi Oct 7 '20 at 3:01