Talk:Grothendieck topology

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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 . 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)

Contents 1 fppf and quasi-finiteness 2 "Comparable" vs. "equivalent" 3 many other cohomology theories 4 Morphisms of sites

[edit] 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)

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)

[edit] "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)

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)

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)

[edit] 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)

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)

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)

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)

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)

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

[edit] 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)