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The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about stacks I would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are particularly boring in this respect, and topological spaces are not much better)

What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology - is there a characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?

And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes which are Noetherian, smooth or even just varieties.

Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.

Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's Foundations of Algebraic Geometry, but I'm after the distilled essence.

PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.


EDIT: I'm not merely after examples of Grothendieck topologies on $Sch$, even though that is handy. I want a reference, if there is one, or just a straight-out answer, that compares the various topologies on $Sch$, and under which circumstances (restricting $Sch$ to a subcategory) they coincide.

For example, does an fppf cover of a variety have local sections over an etale cover? Do the fppf and fpqc topologies give rise to the same sheaves over a nicely behaved scheme? Is the etale topology strictly 'weaker' than some other topology no matter what schemes one looks at? Does one get the same Deligne-Mumford stacks for topology A and topology B?

(Grumble over)


Figure 1 on page 7 of these notes gives a few more of the less common (pre)topologies: cdp, fps$\ell'$ etc.

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A common hierarchy is fpqc --> fppf --> syntomic --> etale --> Nisnevich --> Zariski. There are also the infinitesimal/crystalline sites, but these are in a somewhat orthogonal direction (and one can superimpose e.g. the etale topology on the crystalline site). –  Emerton Sep 5 '11 at 3:01
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Ah, but surely the lattice of topologies is more interesting than some linear order... And I don't get when you say 'etale topology on crystalline site'. To me, saying the blah site means schemes with the blah topology. I'm not an algebraic geometer, so please bear with me. :) –  David Roberts Sep 5 '11 at 4:21
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Emerton, I share David's lack of understanding of the grammar of "etale topology on the crystalline site". In the definition I know, a site is a category equipped with a (Grothendieck) topology. So if S is a site then the phrase "topology on S" isn't one I can parse, unless it just means topology on the underlying category C of S; but in that case you might as well say "topology on C". What am I missing? –  Tom Leinster Sep 5 '11 at 4:42
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What Emerton means by that phrase is (probably) some version the following: 'crystalline site' over $S$ means the category of divided power thickenings of $S$-schemes (over some other base, usually something like $\mathbb{Z}_p$). This is of course not a site till we decree what the coverings are. There are various possibilities, among those being the choice of etale coverings of the thickenings. –  Keerthi Madapusi Pera Sep 5 '11 at 5:02
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There are also the various flavors of the h-topology on Sch introduced by Voevodsky. See Friedlander's notes on them here: math.northwestern.edu/~eric/lectures/ihp/ihplec2.pdf –  Keerthi Madapusi Pera Sep 6 '11 at 1:40

5 Answers 5

I have just discovered a chart comparing topologies on Sch/S, made by Pieter Belmans. It includes all the topologies discussed above, and some more I haven't even heard of. It's even interactive and includes definitions and other information.

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I am thankful for being mentioned, but please observe that I am certainly not satisfied with the current state of the thing: see the accompanying blogpost at pbelmans.wordpress.com/2014/08/01/comparison-of-topologies where I list some of the things I still intend to do. It is by the way exactly this MO question that made me want to do this comparison. –  pbelmans Nov 11 at 16:54
    
Hi Pieter, I guess I should have waited for you to finish it and post it here yourself then! Still, I find it an invaluable resource already, so thanks a lot :) –  Adeel Nov 11 at 17:39

Here is one small point in answer to this question:

The étale site and the fppf site have the same algebraic stacks.

Here I'm saying 'algebraic stack' for a stack of groupoids with a representable smooth surjection from a scheme.

If we further restrict to algebraic stacks with quasi-affine diagonal, then we have:

The étale site and the fpqc site have the same algebraic stacks of this sort.

I learned this from notes on stacks by Anatoly Preygel.


Some more data points from the Stacks Project:

  • smooth covers can be refined by etale covers (tag 055V)
  • tag 02LH could possibly also be useful, if we can identify what sort of map $\coprod_{i=1}^n T_i \to T \to S$ is:

    Let $f : U \to S$ be a surjective etale morphism of affine schemes. There exists a surjective, finite locally free morphism $\pi : T \to S$ and a finite open covering $T = T_1 \cup \ldots \cup T_n$ such that each $T_i \to S$ factors through $U \to S$. if

  • Given an algebraic stack $X$ (defined as in Stacks Project), we can find a presentation by a smooth groupoid $R\rightrightarrows U$ in algebraic spaces (i.e. the source and target are smooth, and we have an equivalence $[U/R] \to X$). Tag 04T5 tells us that $X$ is also equivalent to the stack $[U'/R']$ where $R' \rightrightarrows U'$ is a presentation with source and target flat and locally of finite presentation (so there is a surjection which is flat and locally of finite presentation $U' \to X$).

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Another partial answer.

To quote wikipedia's page on Nisnevich topology:

  • The cdh topology allows proper birational morphisms as coverings.
  • The qfh topology allows De Jong's alterations as coverings.
  • The l′ topology allows morphisms as in the conclusion of Gabber's local uniformization theorem.

The cdh and l′ topologies are incomparable with the étale topology, and the qfh topology is finer than the étale topology.

Of course, can we compare the cdh and l' topologies with other topologies?

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This is separate to the other answer, because it deals with a different aspect of my question. –  David Roberts Oct 3 '11 at 5:02

``For example, does an fppf cover of a variety have local sections over an etale cover?'' (I realize the question is a "big picture" one so that to focus on one small aspect is to miss the point, but I do so anyway.) No. Take a morphism $f:S\to C$ with $S$ a surface and $C$ a curve, both $S$ and $C$ smooth over an algebraically closed field $k$. Assume that the geometric generic fiber is singular; there exist such in positive characteristic. Say $D\subset S$ is the locus of singularities; this is a curve, inseparable over $C$. Now localize: strictly henselize $C$ at a closed point $P$ and strictly henselize $S$ at a point on $D$ over $P$. We now have $S'\to C'$, an fppf cover of $C'$. If there were a section over an etale cover of $C'$ then there would be a section, say $E$. This is a curve in $S'$ having intersection multiplicity $1$ with the generic fiber, while it meets the fiber over $P$ with multiplicity at least $2$, contradiction.

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Or, for a simpler example, over a base ring k, consider A^1->A^1 by x->y^n for n>1. This morphism is finite free of rank n and thus fppf. Let p:S->A^1 be etale. Then \Omega^1_{S/k} is free on p^*dx. If S->A^1 lifts through A^1->A^1 as q:S->A^1, then p^*dx = q^*(ny^{n-1}dy). Thus ny^{n-1} (and hence y, as n>1) pulls back to a unit on S. Since p^*x = q^*(y^n), we see that x pulls back to a unit on S, and so p misses the subscheme (x=0) of A^1. –  user2490 Sep 9 '11 at 15:17

The basic answer is essentially as Emerton described in the comment. The most commonly used topologies on schemes are Zariski, Nisnevich, étale, smooth, syntomic, fppf, and fpqc, and this list is totally ordered by increasing fineness. The canonical topology is finer than the fpqc topology, but I have never seen it explicitly used. You can see a discussion of these topologies (other than Nisnevich) in the Stacks project chapter on Topologies on Schemes.

You ask about restricting to subcategories of schemes to get equivalent topologies, but I think you would have to take unusually small subcategories. For example, the étale and Nisnevich topologies coincide on the spectra of fields only when the fields are separably closed. I think if the Nisnevich covers of a scheme are Zariski covers, then the scheme is zero dimensional. Smooth and étale covers coincide if you restrict to say, varieties of a single fixed dimension. I think the same is true for syntomic versus smooth and fppf versus symtomic (but I am far from sure). If you restrict your schemes to be locally finitely presented over a fixed base, then fppf and fpqc coincide. Even though the étale and smooth topologies are usually not equivalent, they give rise to equivalent categories of sheaves, because every smooth cover has an étale refinement.

The Stacks project has a list of properties that different topologies satisfy, in the Descent chapter. Bjorn Poonen also has a table of permanence properties in Appendix C of his notes on Rational points on varieties.

If you're really hoping for a more interesting looking partially ordered set of topologies, you may consider more exotic examples like the cdh topology (finer than Nisnevich, incomparable with étale), and the naïve fpqc topology, whose covers are faithfully flat quasi-compact maps (incomparable with most of the list). The latter is typically only used by people when they are making mistakes.

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"The latter is typically only used by people when they are making mistakes." :-)) –  DamienC Oct 3 '11 at 9:55

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