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In a recent MO question I asked about the relation between surjective submersions (in the category of smooth or otherwise manifolds) and maps that admit local sections. The latter, it turns out, are more general than the former, as surjective submersions $f:X\to Y$ admit local sections through each point of the domain. This means that for each $y\in Y$ there is an $x\in X$ such that $f(x)=y$ and there is an open neighbourhood $U\ni y$ and a map $\sigma:U\to X$ such that $f\circ \sigma$ is the inclusion of $U$ into $Y$ and $x=\sigma(y)$.

Now there is a class of local-section-admitting maps in $Top$ which have a similar property: every point in the codomain has a local section through it; these are called (surjective) topological submersions. In some respects these are better than just maps which have local sections, because a very complicated map could have very few local sections, none of which pass through regions of interest. Having many local sections seems to force the map to be rather nice. For example, a topological submersion with discrete fibres is a local homeomorphism, whereas a map with local sections with discrete fibres can be pretty wild. Ditto submersions in Diff: such a thing with discrete fibres is a local diffeomorphism.

Now for a general Grothendieck pretopology $J$ on a concrete category $C$ one could ask formally for this sort of map. Define a surjective $C$-submersion to be a map $f:X\to Y$ in $C$ such that for each point $x$ of $X$ there is a covering family $(U_i \to Y)_{i\in I}$ from $J$ such that $x$ is in the image of some family of maps $\sigma_i:U_i \to X$ which are local sections of $f$ (or equivalently, some map $U_i \to X$ for some $i$ which is a local section of $f$).

I can imagine an extension beyond concrete categories, and so here's the question(s) (and I hope it justifies the alg.geom. tag):

(1)Is this sort of setup seen in other categories? (most notably in Schemes (or some subcategory) or a topos)

(2) Since in Schemes (or a subcategory) one has $R$-points for arbitrary (nice enough) rings, one can consider how the collection of local sections changes under extension/restriction of scalars. Is this sort of thing considered?

(3) Do we recover some sort of characterisation of etale maps (or similar) by analogy with the result for Diff- and Top-submersions with discrete fibres?

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  • $\begingroup$ Consider the affine line $X$ over $Y = {\rm{Spec}}(k)$ for an imperfect field $k$. What do you propose to do for closed points $x \in X$ such that $k(x)/k$ is not separable? Demanding sections through all points of the sources seems much too strong if one wishes to work with the etale topology. (The analogy with manifolds misses some aspects.) In practice it is very often used that smooth maps admit etale-local sections, and fppf maps admit quasi-finite flat sections, but neither can be expected to pass through an arbitrary point in a fiber in general. $\endgroup$
    – BCnrd
    Commented Sep 29, 2010 at 8:57
  • $\begingroup$ I'm a complete novice when it comes to alg.geom., so this is just the sort of information I am after. A short (but unsatisfactory) answer would then be 'no, this doesn't generalise from manifolds to schemes', but I presume that there are nice situations when something like this idea has merit? $\endgroup$
    – David Roberts ♦
    Commented Sep 29, 2010 at 9:28
  • $\begingroup$ Dear @DavidRoberts, the map $\mathbb R\overset{x\mapsto x^2}{\longrightarrow}[0,\infty)$ is continuous with discrete fibers yet not a local homeomorphism about $0\in \mathbb R$. Why isn't it a topological submersion? It seems the only problem is $0\in \mathbb R$, but I think we can take the positive branch of the square root as a section. Same for the projection from the vanishing locus $(x-y)(x+y)=0$ onto the $x$-axis. (Sorry if I'm missing something silly.) $\endgroup$
    – Arrow
    Commented Jan 16, 2019 at 14:32
  • $\begingroup$ @Arrow they indeed look to me to be topological submersions! $\endgroup$
    – David Roberts ♦
    Commented Jan 16, 2019 at 21:39
  • $\begingroup$ Dear @DavidRoberts, but then these are topological submersions with discrete fibers which are not local homeomorphisms, contrary to the assertion at the end of the second paragraph. What am I missing? $\endgroup$
    – Arrow
    Commented Jan 16, 2019 at 21:48

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I think that the property you want is the criterion for smoothness in terms of lifting maps over nilpotent thickenings. This is a way of expressing in algebraic terms the idea in manifold theory of having local sections.

The statement is that $f:X \to S$ is a morphism of schemes, locally of finite presentation, then $f$ is smooth precisely when the following is satisfied: given a map Spec $A \to X$, a square zero thickening Spec $B$ of Spec $A$ (so $B$ surjects onto $A$, and the kernel $I$ satisfies $I^2 = 0$), and a map Spec $B \to S$ making the obvious diagram commute, then we can lift the latter map to a map Spec $B \to X$.

If you look at Ex. II.8.6 in Hartshorne it discusses this somewhat obliquely. The book Neron models discusses it more clearly. You will probably have to think quite a bit yourself to really understand the analogy.

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