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There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective on points, such that each map in the family has some property $X$ characteristic of the topology in question; the topology is generally then called the "$X$ topology". For instance, in the étale topology, each of the maps is required to be étale.

The properties $X$ in question vary widely, but here's one very basic feature which many of them have in common: very often "$X$" is of the form "$Y + Z$", where $Y$ is a "niceness" property and $Z$ is a "finiteness" property. For instance, "etale" decomposes as "formally etale" + "locally of finite presentation". My question is:

Question 1: Why do topologies used in algebraic geometry generally involve specific finiteness conditions?

I think the answer to this question may well be quite straightforward -- for instance, I'm not sure whether a "formally etale" topology exists, but if it does, I imagine it has rather pathological properties, and it would be interesting to know something about these pathologies.

Question 2: Are there counterexamples to this trend in algebraic geometry -- useful topologies which allow for "non-finite" morphisms to generate their covers?

Question 3: Are there counterexamples to this trend elsewhere in mathematics -- useful Grothendieck topologies which are more "infinitary" in nature?

For Question 3, I'm kind of suspecting that some of the topologies used in set-theoretical forcing might be more "infinitary" in nature, but I'm having trouble imagining less exotic settings where this sort of thing is likely to come up.

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    $\begingroup$ This seems quite philosophical, but I am not sure whether there is any finiteness in the pro-étale topology, and I don't see any finiteness in quasisyntomc / fpqc / $v$-topology. $\endgroup$
    – Z. M
    Nov 22 at 16:50
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    $\begingroup$ My knowledge of algebraic geometry is too superficial to answer with certainty, but can't this be explained by the fact that the important site is the one with only finitely presented rings (because that's what classifying topos gives) where the finitness condition are automatic, and then one embeds all scheme in the resulting topos and take the indices topology on this ? $\endgroup$ Nov 22 at 16:52
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    $\begingroup$ @Z.M even in the fpqc or pro-étale topology, although the underlying objects are not of finite type, there is always a running coherence assumption on the covers. Morally, affine schemes should form a basis for any "geometric" Grothendieck topology, and covers of affines by affines should be generated by finite covers. This means that you can reduce questions to commutative algebra in many cases. (Note that an infinite disjoint union of nonempty affine schemes is not affine, so that's the sort of thing you're trying to avoid.) $\endgroup$ Nov 22 at 21:04
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    $\begingroup$ A partial philosophical explanation for the ubiquity of finite presentation hypotheses in algebraic geometry is that the origin of the area is the study of definable sets in the theory of commutative rings (or even algebraically closed fields). So finiteness is forced [no pun intended] by the model theory. $\endgroup$ Nov 22 at 21:08
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    $\begingroup$ I want to second Z. M's comment: There's been some trend to just omit those finiteness hypothesis (except for a very mild one guaranteeing quasicompactness of affine), e.g. fpqc, pro-etale, v-topology etc.; also condensed sets can be seen as an incarnation of this push to omit finiteness hypothesis on the morphisms. Little is lost by doing so, but one gains the ability to localize a lot more, and make the space pretty simple (e.g., schemes all of whose connected components are the spectrum of a valuation ring with algebraically closed fraction field). $\endgroup$ 2 days ago

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I guess a conceptual explanation is that algebraic geometry deals with localizations in order to glue global from local data, but the functor $M \mapsto M[f^{-1}]$ only preserves finite limits. In fact, $M[f^{-1}]$ is a filtered colimit of copies of $M$, so this is an instance of "filtered colimits commute with finite limits".

But I also doubt that, for example, the formally étale morphisms produce a subcanonical topology, which is what we want for sure.

Isn't the answer to Question 3 (Grothendieck topologies which are "infinitary") any non-compact topological space?

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Finite presentation assumptions in algebraic geometry are usually there because of Noetherian approximation. Namely, if $f : X \to Y$ is a morphism of finite presentation and $Y = \operatorname{lim}_I Y_i$ is a directed inverse limit, then there exists an index $0 \in I$ and a morphism $f_0 : X_0 \to Y_0$ such that $X \cong X_0 \times_{Y_0} Y$ as schemes over $Y$. This is particularly useful because if $Y$ is qcqs then it can be written as such a limit of Noetherian schemes (note the properties this question is concerned about are local on the target so we can usually assume $Y$ is affine).

The point is that finite presentation morphisms inherit many of the good properties that finite type morphisms of Noetherian schemes enjoy. For example, formally smooth maps of Noetherian rings are automatically flat which by approximation implies that formally smooth (and in particular formally étale) maps which are locally of finite presentation are flat. Approximation also implies that formally étale and locally of finite presentation is equivalent to $X$ locally being the vanishing of a polynomial with nonvanishing derivative (so-called standard étale).

Here is an example of the type of pathology that can occur without the finite presentation hypothesis. Let $0 \neq I \subset R$ be an ideal with $I^2 = I$ (note this forces $I$ to be infinitely generated). Then the ring map $R \to R/I$ is formally étale but certainly not flat. As Martin Brandenburg pointed out, there is no reason for such non-flat maps to generate a subcanonical topology.

Of course one could consider a topology generated by flat and formally étale maps which are not necessarily locally of finite presentation. One issue with this topology is just that this is probably a difficult condition to work with for maps that are not locally of finite presentation. One could hope that any flat formally étale ring homomorphism $R \to S$ is the filtered colimit of étale ring homomorphisms but this is probably false (see this answer for a counterexample when étale is replaced by smooth).

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    $\begingroup$ Gabber has an example of an imperfect $\mathbb F_p$-algebra (hence not ind-étale) with trivial cotangent complex. $\endgroup$
    – Z. M
    Nov 23 at 7:33
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To give an interesting data point, I can offer two examples of something that seems a bit more infinitary.

If we consider the category of finite-dimensional manifolds, and covers being submersions (or perhaps families of jointly-surjective submersions), then we have a kind of finiteness condition, namely that the fibres are also finite-dimensional manifolds.

But one can put finite-dimensional manifolds into the larger category of Fréchet manifolds, which can be rather infinite-dimensional. Then submersions are defined in such a way so that one has local charts that are split. Now here surjective submersions (or jointly-surjective families of submersions) have fibres that are very much without any kind of finiteness condition. Finite-dimensional manifolds are, however, paracompact, and often taken to be second-countable (though it's really usually unnecessary). So an arbitrary jointly-surjective family can be shuffled slightly to not be so radically infinite. But if one takes an arbitrary Fréchet manifold, and consider a jointly-surjective family of submersions, then there is very little one can consider as a kind of cardinality bound.

To get even more radical, one can consider the category of diffeological spaces, and where covers are subductions. Every diffeological space has a canonical cover given by the copairing of all possible plots. This is a very large cover. And, for example, one can take an infinite-dimensional Fréchet manfiold and then as all the plots have finite-dimensional domains, there's no way to refine this cover by anything that's very much smaller. But, perhaps since the finiteness in the algebraic geometry setting is a relative concept, one might be able to figure out a way that looks like some kind of cardinality bound on fibres.

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