Talk:Grothendieck topology

I'm not an expert, but I'd suggest to replace the term 'open immersion' with 'inclusion', since the former one is used rather in the context of manifolds.

Please sign your posts, Mr. 128.100.216.221. Use four tildes (the ~ thingy).

I wrote "open immersion" because I come out of the world of algebraic geometry. An open immersion in algebraic geometry is the inclusion map from an open subset U of an algebraic variety to the entire variety, and it's the correct term in that context; but I didn't notice that the term was confusing in the context of manifolds! I'm not convinced, however, that it would be appropriate to change "open immersion" to "inclusion" everywhere. I'm not sure what use Grothendieck topologies have for people interested purely in manifolds; so far as I'm aware, in practice they only turn up via schemes or rigid analytic spaces. Manifolds already come with good (topological space-style) topologies, and so do spaces of functions like

C^{\infty }(M)

. Only crazy people like me talk about Grothendieck topologies, and for us, "open immersion" is the correct term.

For the moment, I've added a short clarification to the article. If that's not sufficient, please edit or make a suggestion. 141.211.62.20 01:55, 29 January 2007 (UTC)[reply]

fppf and quasi-finiteness

Is the definition of the fppf topology right? Every faithfully flat finitely presented morphism has a quasi-finite refinement, but I would have thought all faithfully flat and finitely presented morphisms should be considered covers. Changbao 10:16, 10 February 2007 (UTC)[reply]

Changbao,

I'm going off of what I read in SGA 3, see [1]. However, Vistoli defines fppf as you suggest in [2], page 29. Honestly I don't know very much about flat topologies, so I can't answer your question in a meaningful way. 141.211.63.85 00:34, 11 February 2007 (UTC)[reply]

"Comparable" vs. "equivalent"

I have reverted the sentence "Grothendieck topologies are not equivalent to the classical notion of topological spaces." to "Grothendieck topologies are not comparable to the classical notion of topological spaces." My reasoning is as follows: Both sentences state that the two concepts are inequivalent, but only the latter indicates that neither concept is a generalization of the other. Old versions of the article hinted that Grothendieck topologies were more general, but this is false for pathological topological spaces, and I don't want anyone to come away from this article with that misconception. 141.211.62.20 00:49, 13 February 2007 (UTC)[reply]

I don't like "comparable" because, under its plain meaning, the theory of Grothendieck topologies exists to be compared to classical topology. What do you think of my rewrite? Changbao 06:44, 13 February 2007 (UTC)[reply]

Very good, better than before. I copyedited it a little—I thought it was important to mention that topological spaces give sites, not just Grothendieck topologies, since I thought that might confuse a beginner. Also, I wouldn't call the differences between the two theories minor. Yes, for sober topological spaces they're the same, but the setup for the theories is completely different, and there are lots of useful Grothendieck topologies which don't come from topological spaces, such as the étale and flat sites. Feel free to refine the paragraph even more. 141.211.62.20 00:32, 14 February 2007 (UTC)[reply]

BTW why the indiscrete topology does not come from a site? —Preceding unsigned comment added by 128.100.216.180 (talk) 23:31, 5 March 2009 (UTC)[reply]

It's not a sober space (if it has more than one point). For, suppose that x and y are points of the space. Then the closure of either x or y is the intersection of all closed set containing them; and the only non-empty closed set is the whole space. So both x and y are generic points of the space. Ozob (talk) 16:24, 6 March 2009 (UTC)[reply]

many other cohomology theories

The following sentence is misleading:

It has been used to define many other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology.

My understanding is that l-adic cohomology is exactly the same as etale cohomology (and currently, that's supported by linking on Wikipedia). This leaves us with only two new examples, flat and crystalline. I am not familiar with the former (and there is currently no article on it), but is two sufficient to claim many other? Are there more examples that can be quoted? If not, it would be best to temper the statement a bit. Arcfrk 23:11, 27 March 2007 (UTC)[reply]

By l-adic cohomology one usually means just the étale theory with l-adic coefficients, certainly. (An infinite number!) Flat cohomology is two theories. The word 'many' can be removed.

By the way, I would like to create flat cohomology as an article, by moving out the definition of the flat site(s) from this one. Only from a sort of fundamentalist Grothendieck position is it an example that one needs early on. The applications have to be cited to justify it, really. (I wonder if there is a book on it? There are books on the étale and crystalline theories. Quite indicative.) Charles Matthews 21:33, 3 April 2007 (UTC)[reply]

I think the logic of moving flat cohomology to a separate article is not convincing: it implicitly assumes that these topologies exist only to provide us with cohomology theories. However, they of course are also a basic ingredient of descent theory, and from this viewpoint the flat cohomology is arguably even the most central. Indeed, flat topology being "close" to the canonical topology, a functor being a sheaf in pfqc-topology is a much stronger implication of being potentially representable than being sheaf in, say, étale topology. Thus I think there is ample reason for introducing the flat topology here in the main article. Stca74 18:20, 10 September 2007 (UTC)[reply]

This is a very good point. Descent is hugely important. But once we include the two flat sites, the Grothendieck topology article is pretty long and spends a rather large amount of time talking about schemes, which are, from the general perspective, rather special. And if we're going to include all the interesting sites, then we'd need to include the crystalline site (which is probably worth including anyway, since it doesn't have final objects), and then the article would be very long and very unfocused.

I think a better solution is to break off this section of the article into a new article, Topologies and schemes. This could include all the topologies that get used in practice (including, for instance, SGA 3's finite étale topology and Suslin-Voevodsky's h and qfh topologies (these turn up in the homology of schemes; they're supercanonical, so very weird.)). Ideally it would also have some of their properties; so, the Zariski topology section could say that all constant sheaves have vanishing higher cohomology; the Nisnevich topology section could mention its use in algebraic K-theory; and the fpqc topology section could explain why it's relevant to descent.

I propose that, unless someone has an objection, that someone cuts and pastes to create Topologies of schemes. What does everyone else think? 141.211.63.198 02:38, 30 September 2007 (UTC)[reply]

I think that very technically speaking this isn't right. Etale cohomology with coefficients in the abelian group Q_l yields garbage. You need to take account of the natural topology on Q_l somehow, usually in two steps: 1. take the inverse limit over finite coefficients and 2. tensor the resulting Z_l-module with Q_l. (Not that this has anything to do with the article.) Changbao 22:54, 3 April 2007 (UTC)[reply]

Yes, that is explained in the article l-adic cohomology. Charles Matthews 19:28, 8 April 2007 (UTC)[reply]

Morphisms of sites

Some (other) anonymous user tried to define a morphism of sites. The definition used, though obvious, is incorrect, so I've removed it. The correct notions are continuity [3] and cocontinuity [4]. [5] gives an example of a functor between two sites which preserves covering but which is not continuous, i.e., does not preserve sheaves, and that's the condition which is really useful. 141.211.120.63 19:11, 8 June 2007 (UTC)[reply]

Zar is not the restriction of Spc

In the example given in the article (Spec A -> Spec A/N), as "underlying topological spaces" you mean the spaces given by the closed point? Since if you take the whole spaces, I don't think that Spec A $\rightarrow$ Spec A/N is always a topological open immersion (Spec $\mathbb {C} [x]/(x)\rightarrow$ Spec $\mathbb {C} [x]/(x^{2})$ is a closed immersion even topologically). —Preceding unsigned comment added by 87.8.176.101 (talk) 10:14, 14 September 2008 (UTC)[reply]

I think the article is correct here. You got the direction of a map mixed up, which may be the source of confusion: A → A/N is the quotient by an ideal, so it induces a map of spectra in the opposite direction: Spec A/N → Spec A. This map of spectra is always a homeomorphism of topological spaces. (It's a bijection because N is contained in all prime ideals, and by checking on a basis for the topology you can see that it's a homeomorphism.) The map Spec C = Spec C[x]/(x) → Spec C[x]/(x²) is an example of this; it is induced by a map of rings C[x]/(x²) → C which is the quotient by the nilradical, which is the ideal (x).

The point of the article's statement is that even though this homeomorphism of ring spectra is a topological open immersion, it is not a scheme-theoretic open immersion. It's true that the morphism is both a topological closed immersion and a scheme-theoretic closed immersion, but that's not the point the article is making here. Ozob (talk) 22:36, 14 September 2008 (UTC)[reply]

Definition of a grothendieck pretopology

Why does this page only define Grothendieck pretopologies for categories with pullbacks? Couldn't we get rid of (PT 0) and replace (PT 1) with

(PT 1') If $\{f_{i}:X_{i}\rightarrow X|i\in I\}$ is a covering family of X, then for any morphism $g:Y\rightarrow X$ there is a covering family $\{h_{j}:Y_{j}\rightarrow Y|j\in J\}$ such that for each j, $g\circ h_{j}$ factors through some $f_{i}$ .

Then in the case that C has pullbacks, we can derive (PT 1). When we don't have pullbacks we still get a Grothendieck topology by defining covering sieves to be those which contain some covering family.

Steven Gubkin (talk) 21:26, 14 November 2008 (UTC)[reply]

Yes, you could do that, but it's not standard. (Remember, we don't make innovations here on WP, we just quote what other people have done.) I think also you run into the problem that there's no advantage to doing things the way you're proposing; rather than use (PT 1'), one might as well work with sieves and topologies. Most of the time, people use pretopologies because they find them a little easier, and with (PT 1') you lose some of that. Ozob (talk) 00:42, 15 November 2008 (UTC)[reply]

Comment

Hello there, I found this article as an excellent explanation of the Grothendieck topology subject. I would like to suggest you to add the book of S. MacLane and M. Moerdijk (Sehaves in geometry and logic, Springer) as a reference, I found it quite useful and maybe it would help others too! Keep up the good job fella!

--Dieu reconnaîtra les siens 10:58, 12 February 2009 (UTC) —Preceding unsigned comment added by Kriega (talk • contribs)

Pullback in case of topological space

I have some problems to apply the definition of the pullback in the case of topological spaces:

Let X be a topological space, f:U→V an inclusion of open subsets of X and S a sieve on V, that is, a collection of inclusions g_i:V_i→V of open subsets V_i of V. Then for an open subset W of X the set S(X) consists precisely of g_i if W=V_i and is empty else. The definition of the pullback in the article yields that f^*S(W) consists precisely of h:W→U if and only if W is a subset of U - otherwise Hom(W,U) would be empty - and fh:W→V is in S(W). But S(W) contains at most one element. Hence f^*S selects only those inclusions from S whose domains lie in U. But this is in general not the collection V_i ∩U → U as mentioned in the article: Take, for example the sieve that consists only of V→V and let U be a proper subset of V.

Probably, the fiber product is meant on the level of objects? Thanks for any comments and excuse my poor language.

Best regards, Florian —Preceding unsigned comment added by Florian Geiß (talk • contribs) 15:30, 9 November 2009 (UTC)[reply]

Remember that a sieve S on V is a subfunctor of Hom(—, V). In particular, if S(V_i) is non-empty, then for any subset Z of V_i, S(Z) is also non-empty. For instance, there is no sieve on V such that S(V) = 1_V and S(Z) = ∅ for all Z ≠ V. Ozob (talk) 02:33, 10 November 2009 (UTC)[reply]

Right. I was mislead by the passus In the case of O(X), a sieve S on an open set U corresponds to a collection of open subsets of U "selected" by S. Maybe it would help to add some information that not any family occurs as a sieve in this example. Thanks anyway. Florian, 2009-11-10

Motivation

Looking over the paragraph 'Motivation' in the article, the phrase (beginning the second-to-last sentence)

"Because the value of a presheaf on an open set determines the value on any smaller set by restriction"

seems misleading - it may be misread as the statement that all sheaves are flasque. Clearly a section will be described over smaller sets by the restriction mappings, but it is not clear to me that this sentence refers to sections and not the entire set/module/algebra of sections over an open set.

I think that 'value of a presheaf' is the culprit, as this makes me think 'set/module/algebra/whatever of sections' and not 'specific section'. (Hopefully I am not just deranged.) Could it be clearer to state something like

"Given a section s over an open set U and an open V/subset U, the restriction map describes what s does 'at V'"

and avoid the implication that restriction maps should be surjective?

(It seems unusual that the sheaf idea is being 'hidden' in a page on Grothendieck topologies, but if that is important then I'll try something else.)

JBroll (talk) 08:19, 26 April 2010 (UTC)[reply]

I've rewritten that part; I agree with you that it was unclear, but I took a different approach to fixing it than the one that you suggested.

I don't think it's too odd that sheaves are discussed here. They may be better off at topos, but this isn't bad. I don't think sheaf (mathematics) is the right place because sheaves on topological spaces can be discussed in a much more elementary way than sheaves on sites. Ozob (talk) 03:50, 27 April 2010 (UTC)[reply]

"Coarsest to finest" order not pratical

Hi, I would like to add a few new examples of Grothendieck topologies, namely those coming from the study of motives and A^1-homotopy theory. I have already added remarks about variants of the Nisnevich topology. However these do not fit into a "coarsest to finest" order but rather a poset. I will try to include a graphic representation explaining the relationships between those. Do you think this should go into the examples section or rather another paragraph (as those examples are rather more exotic than Zariski, étale...) ? —Preceding unsigned comment added by SimonPL (talk • contribs) 20:41, 7 August 2010 (UTC)[reply]

Well, this section has annoyed me for some time because it dominates the article: A huge part of the article is taken up by this enormous list, most of which is of no interest to someone learning about Grothendieck topologies for the first time. So I've separated things out into a bunch of new articles. (Some of these existed before but were redirects to here.) We now have:

as well as redirects

I think this is a better arrangement. But of course everyone is free to improve, modify, ruthlessly criticize, etc. my work. In particular, fixes from experts are welcome. (One particular thing that needs improvement: I suspect l′ topology should be its own article, but I've never encountered it before, so I'm not qualified to write it.) Ozob (talk) 18:55, 8 August 2010 (UTC)[reply]

(T 1) explanation

I think the explanation of (T 1) goes the wrong way: The maximal sieve is the one generated by the minimal cover, so (T 1) corresponds rather to the idea that each set covers itself. — Preceding unsigned comment added by 82.20.208.189 (talk) 17:48, 11 February 2018 (UTC)[reply]

Integral?

The "integral" word links to "integer". Shouldn't it be "integer coefficients" rather than "integral coefficients"? TricksterWolf (talk) 12:41, 12 February 2021 (UTC)[reply]