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

Question

Also at MO.

---

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?

Answer 1

Let me just answer this to get off the unanswered list.

It is somewhat true that one can develop a fairly rich theory of formally smooth maps without the assumption of being locally of finite presentation. For example, the whole contents of [1, Chapter 2] discuss precisely this. You get, for example, versions of the Jacobian criterion (correctly interpretted) and relationships with the notion of geometric regularity.

But, there is some essential 'glue' that, in my opinion, is missing even if you don't care about the intuition of your maps having fibers that look like 'smooth finite-dimensional manifolds'.

Before we explain, let us give some examples of formally smooth maps:

1. (Example 1) Any smooth morphism $X\to Y$ is formally smooth--this is the infinitesimal lifting criterion (e.g. see [2, Tag02H6])
2. (Example 2) Let $X$ be a scheme and $x$ a point of $X$. Then, the map $\mathrm{Spec}(\mathcal{O}_{X,x})\to X$ is formally smooth-- this follows from [1, Example 2.2.2(b)].
3. (Example 3) Let $K$ be a field and $L$ a separable extension of $K$. Then, $\mathrm{Spec}(L)\to\mathrm{Spec}(K)$ is formally smooth--this follows from [1, Corollary 2.4.6].
4. (Example 4) For any separable field extension $L_1$ and $L_2$ of a field $K$, then the morphism $\mathrm{Spec}(L_1\otimes_k L_2)\to \mathrm{Spec}(L_i)$ is formally smooth for each $i$--this follows from Example 3 and [2, Tag02H2].
5. (Example 5) Let $X\to Y$ be a flat closed embedding, then $X\to Y$ is formally smooth--this follows from [2, Tag04FF] and [2, Tag02GC]. There exist flat closed embeddings of the form $\mathrm{Spec}(A/I)\to\mathrm{Spec}(A)$ where $\mathrm{Spec}(A)$ is connected--see [3].

Now, obviously Example 1 is geometrically what we want intuitively. The Example 2 definitely looks weird from our usual picture of what smooth maps look like but, it kind of is supposed to be imagined as an 'infinitely small Zariski neighborhood around $x$' which seems reasonable enough to call smooth. The Example 3 looks really weird but, you know, 'arithmetic stuff--what are you going to do?' Example 4 is starting to make you sweat because tensor products of fields can be heinous (e.g. see [4]). Example 5 should send you reeling--what unholy monstrosity have you unleased upon this accursed land to make that map be 'smooth'?

But, let's highlight three of the properties that this coterie of fiends in Example 2 and Example 5 break that we expect from smooth morphisms. Namely, we have the following well-known results:

Theorem 1 ([2, Tag056G]): Let $f:X\to Y$ be a smooth morphism. Then, $f$ is an open map.

Theorem 2 (Grothendieck's theorem, [2, Tag025G]): Let $f:X\to Y$ be universally injective and étale. Then, $f$ is an open embedding.

We note then that our devlish friends all violate the conclusions at least one of these theorems:

Example 2 issues: The map $\mathrm{Spec}(\mathcal{O}_{X,x})\to X$ violates the conclusions of Theorem 1 and Theorem 2 in most reasonable situations. For example, if we assume that $X$ in Example 1 is integral and finite type over some field $k$ then the map $\mathrm{Spec}(\mathcal{O}_{X,x})\to X$ has image $U$ which is not open. Indeed, if this were the case then the image would be an open subset of $X$ containing at most $1$ closed point (depending on whether $x$ is closed or not) which is not possible for dimension reasons since if $d=\dim(X)=\dim(U)$ then by Noether normalization we have a surjective map $U\to \mathbb{A}^d_k$ which shows the existence of infinitely many closed points. We also see that when the conclusion of Theorem 1 doesn't hold, then neither does the conclusion of Theorem 2 since $\mathrm{Spec}(\mathcal{O}_{X,x})\to X$ is universally injective (e.g. the map is evidently radiciel in the sense of [2, Tag01S3] and one can then apply [2, Tag01S4]).

Example 5 issues: Again, if $X\to Y$ is as in Example 5 and $Y$ is connected then it violates the conclusions of Theorem 1 and Theorem 2. Indeed, since $X\to Y$ is a closed embedding with $Y$ connected we know that its image is not open which violates the conditions of Theorem 1. The same idea as in our discussion of the issues with Example 2 show that the conclusion of Theorem 2 are also violated.

References:

[1] Majadas, J. and Rodicio, A.G., 2010. Smoothness, regularity and complete intersection (Vol. 373). Cambridge University Press.

[2] Various authors, 2020. Stacks project. https://stacks.math.columbia.edu/

[3] https://mathoverflow.net/a/227/38867

[4] https://mathoverflow.net/a/352511/38867

[5] Fu, L., 2011. Etale cohomology theory (Vol. 13). World Scientific.