Since the last update about a week ago we have started adding some very basic material on algebraic stacks. It should be possible to lay out the basics of theory of algebraic stacks in the rest of the summer. Mainly I mean material on types of morphisms of algebraic stacks (how to define them, how do you check for a property, how are they related, etc), perhaps a little bit on quasi-coherent sheaves on algebraic stacks, more examples of algebraic stacks, etc. It will probably turn out that during this process the existing material on algebraic spaces will have to be expanded upon and improved.
What are some more exciting topics that I hope to get to in the not too distant future? Here are some possibilities:
- Artin’s axioms, starting with a chapter on Schlessinger’s paper.
- Olsson’s paper on the equivalence of the different definitions on proper.
- Translating some of the more interesting results on algebraic spaces (regarding points, decent spaces, etc) into results on algebraic stacks.
- A proof of Artin’s trick for algebraic stacks to prove that [U/R] is an algebraic stack if s, t : R —> U are flat and locally of finite presentation.
- Diagonal unramified <=> DM
- Prove the correct version of the Keel-Mori theorem in our setting.
- Discuss the lisse-etale site (I may somehow want to avoid this — perhaps you can always avoid working with this beast?)
- Treatment of simplicial schemes (as an example of simplicial objects) and relation to algebraic stacks. Formulate possible extension to higher algebraic stacks (e.g., hands on version of algebraic 2-stacks with example moduli of stacky curves)
- Gabber’s lemma and more on finding nice maps from schemes to algebraic stacks.
- Tricks with limits. Quasi-coherent sheaves are directed colimits of finite type quasi-coherent sheaves if X is…
- Gerbs and relation to H^2.
- Stacky curves. What are 1-dimensional stacks like?
- General Neron desingularization (needed for 1).
Of course it will take a long time to add all of these topics. But it probably is a good idea to take one or two of these topics and keep them in mind while developing the general theory, so as to have a good reason for building theory. Moreover, many of these topics require additional material on schemes and algebraic spaces. For example, 4 requires a bit about Hilbert spaces (only relative dimension 0 however).
Finally two questions:
- A while back David Rydh suggested we give a name to algebraic stacks with locally quasi-finite diagonal. Should we? If so any suggestions?
- And what to call an algebraic stack with finite diagonal? (Maybe these should have been called separated and the ones with only proper diagonal something else. Too late to change now since it has been used like this for a while now in the literature.)
Maybe we could call an algebraic stack with locally quasi-finite diagonal “space-like”? And then the algebraic stacks with finite diagonal are space-like separated algebraic stacks. Hmm… not sure.
Certainly “space-like” is a very poor choice for an adjective describing an algebraic stack whose diagonal is locally quasi-finite. I tried to look for words similar to separated, but “sundered”, “divorced”, or “disunited” do not really sound right either. We could look for a modifier of “separated” and say “weakly separated” but that doesn’t work very well either. I also thought about naming it after a person or persons, but nobody comes to mind who really studied this condition extensively. Arrgh!
How about “granular”? No, that would lead to weird combinations like a “smooth granular algebraic stack”. Another suggestion is “distilled” which may be better. The same example leads to a “smooth distilled algebraic stack” which is still weird, but maybe less so. Also, “distilled” is closer to “separated”.
How about “insulated”?
Actually, that’s not bad, but I think it is still not really right. For now I went with “distilled”… (see current chapter on morphisms of stacks).
How about “cleft? “Cleave” is a synonym for “separate” so “cleft” is a synonym for “separated”.
Aha! Cool. So for example we would get a “smooth cleft algebraic space” and “cleft morphism of algebraic spaces”. Why not “cloven”? What is really the difference between “cleft” and “cloven”?
The same as the difference between ‘sang’ and ‘sung’. Or ‘hung’ and ‘hanged’. Or ‘smote’ and ‘smitten’. Or ‘ate’ and ‘eaten’. ‘Cleft’ is the past tense, ‘cloven’ is the past participle. In this context, it would be ‘cloven algebraic space’ (I’m not sure why the facial disfigurement isn’t ‘cloven palate’).
Also, ‘cleave’ is not a synonym for ‘separate’. Cleaving is something you do with a battle-axe (confusingly, ‘to cleave to’ also means ‘to stick to’). I think ‘split’ is a milder synonym for ‘separate’ (and there’s no past tense/past participle confusion).
Or since the OED says ‘separate’ is from Latin, what about taking the Greek word? Or the German, but ‘getrennt’ sounds weird and I don’t think ‘trennen’ has any English cognates.
Johan, doesn’t the valuative criterion for separatedness of stacks “justify” the definition of separatedness in terms of proper diagonal, at least assuming the diagonal is quasi-compact (i.e., given morphism is quasi-separated)? Anyway, to my ear “cleft” sounds horrible (it also makes me think of the facial physical disfiguration) and “cloven” makes me think of horses (cloven hoof). Oy, please do not encourage these words. How about “quasi-DM” (since after pullback to a scheme mapping to the target stack the pullback is a stack which is almost DM except that isom groups at geometric points are 0-dimensional and lft instead of etale)? Or “pseudo-DM” if “quasi-” is getting overused.
Since finite = locally q-finite + proper for algebraic spaces, for Artin stacks the diagonal (which is rel. rep’tble in alg. spaces) is finite precisely when it is loc. q-f and proper, which is to say the original map is “quasi-DM and separated”. So it seems that there’so no need for a special name for having finite diagonal: it’s just separatedness plus whatever name is chosen for having loc. q-f diagonal.
OK, I did think of “something-DM” I also thought about “KM” since reading the paper by Keel and Mori made me realize that these “distilled” stacks are in some sense just as good as DM stacks. Since your opinion is the strongest one yet, I think I will change “distilled” —> “quasi-DM”. I still do not like it since I do not think it is so nice to have an abbreviation (but a “quasi Deligne-Mumford algebraic stack” is very long). I intend to keep writing Deligne-Mumford algebraic stack in the stacks project, whenever in conversation you would simply say DM stack.
Aha, maybe the solution is the following: In the separation axioms for morphisms I am missing the definition of a morphism f of algebraic stacks whose “fibres” are Deligne-Mumford algebraic stacks, i.e., where Delta_f is unramified. I should introduce this now and call this a “DM morphism”. Then Delta_f being locally quasi-finite will be called “quasi-DM” as you suggest. And an algebraic stack will be called “quasi Deligne-Mumford” if the morphism to Spec(Z) is quasi-DM. OK?
You also get lots of points for pointing out how ridiculous “cleft” and “cloven” are:)
Johan, sounds good to me. (By the way, for some of us KM means Katz-Mazur, so I’m in favor of not using it as shorthand for Keel-Mori. But that now seems moot.)
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