Timeline for Non-Noetherian Stable Homotopy
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
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| Jun 7, 2012 at 17:25 | comment | added | Dylan Wilson | @David: Oh I get it. But yeah, that should be called "stable module category" not "stable derived category". | |
| Jun 7, 2012 at 17:21 | comment | added | Dylan Wilson | What does "stable derived category" even mean? I know of no derived categories that are not stable... And Fernando, apologies- I think we agree completely, I was just confused by the comment "the derived category of a non-noetherian ring is still very much unknown". I guess you were talking about Bousfield classes, etc. Sorry for this silliness. | |
| Jun 7, 2012 at 16:24 | vote | accept | Jonathan Beardsley | ||
| Jun 7, 2012 at 12:16 | comment | added | Fernando Muro | Guys, you're making yourself a mess. Shipley's paper linked above is stable and abelian. | |
| Jun 7, 2012 at 11:26 | comment | added | David White | @Dylan: I haven't thought about Daniel's work in a while, so I hope I don't have this wrong, but he's finding a model structure on the stable module category (mod out by projective modules) over an arbitrary ring $R$. I don't believe this category arises out of a spectrum. It's homotopy category is the stable derived category, so there's a triangulated structure. Doesn't Shipley's work get you an unstable derived category? Anyway, even if you got the same homotopy category I'm sure the model categories are different. In particular, Daniel's is Abelian (constructed using Cotorsion Pairs) | |
| Jun 7, 2012 at 11:21 | comment | added | David White | @Jon: Daniel was my academic brother, so I have lots of notes from his talks. I'll bring them down to Virginia next week and we can chat about it there. | |
| Jun 7, 2012 at 9:09 | comment | added | Fernando Muro | @Dylan you're taking me terribly wrong, I've given a reference above to the theorem saying that $D(R)$ is a stable homotopy theory, i.e. the contrary of (a), and about (b) I've said that most triangulated categories come from stable model categories, since I don't believe Jon is interested incounterexamples. | |
| Jun 7, 2012 at 8:27 | answer | added | Dylan Wilson | timeline score: 4 | |
| Jun 7, 2012 at 7:49 | comment | added | Dylan Wilson | I guess I'm confused as to whether you're claiming that (a) we don't know if D(R) is a stable homotopy category, or (b) we don't know that every stable homotopy category comes from a stable model category. I disagree with (a), as I said above, and (b) is probably true using the same example that you give in your paper on the subject. What have I misunderstood? | |
| Jun 7, 2012 at 7:09 | comment | added | Dylan Wilson | @Fernando: Yes but the homotopy category of a symmetric monoidal stable model category is a stable homotopy category, yeah? For example, HR-module spectra with the smash product, where R can, in particular, be any ring | |
| Jun 7, 2012 at 6:47 | comment | added | Fernando Muro | @Dylan Yes, as I said in my very first comment, most of triangulated categories come from stable model categories, no to be confused with stable homotopy categories in the sense of [HPM]. I feel we're getting muddled up with the terminology here. | |
| Jun 7, 2012 at 4:43 | comment | added | Dylan Wilson | @David: I'm having trouble seeing how Daniel's work is not subsumed in Shipley's work above- what am I missing? | |
| Jun 7, 2012 at 4:41 | comment | added | Dylan Wilson | much much more... | |
| Jun 7, 2012 at 4:41 | comment | added | Dylan Wilson | @Fernando: Here's what's confusing me... we seem to have a monoidal model structure on Ch($\mathcal{O}$) for any locally ringed space, via results in here: ams.org/journals/tran/2006-358-07/S0002-9947-06-04157-2/… Since the homotopy category thereof IS the derived category... that gives the results for most cases of interest, yes? You're right that it doesn't work for an arbitrary Grothendieck abelian category BUT this seems to say that we can do it for, in particular, non-noetherian rings. Also the paper of Shipley certainly takes care of that and | |
| Jun 7, 2012 at 1:32 | comment | added | Jonathan Beardsley | @David that sounds like exactly the type of thing I would like to know about. All in all, it sounds like the answer to my question is: no. There does not seem to be much of a general theory. | |
| Jun 7, 2012 at 0:43 | comment | added | David White | @Fernando's second comment: Daniel Bravo has done some work towards the stable derived category of non-Noetherian rings. In particular, his thesis proves that for any $R$ you have a model category whose homotopy category is the stable derived category. See the following link for more detail. There are a lot of slides, and a preprint is in the works. people.usm.maine.edu/dbravo/publications.html | |
| Jun 7, 2012 at 0:38 | comment | added | David White | Here's a potentially useful source: Mark Hovey's "Open Problems" page. In this link he lists 2 problems for non-Noetherian homotopy theory which may or may not have been answered by now: math.wesleyan.edu/~mhovey/problems/axiomatic.html | |
| Jun 6, 2012 at 23:14 | comment | added | Fernando Muro | @Dylan You cannot check that the derived category of Grothendieck category is a stable homotopy category in the sense of Hovey-Palmieri-Strickland in such generality because you would lack of a tensor structure and some other features. The best result in that direction I know is due to Alonso-Jeremías-Pérez-Vale arxiv.org/pdf/0706.0493.pdf on the basis of previous work by Alonso-Jeremías-Souto | |
| Jun 6, 2012 at 22:40 | comment | added | Dylan Wilson | Alternatively one could use homepages.math.uic.edu/~bshipley/zdga17.pdf and all the hard work of May and others to make stable model category structures! | |
| Jun 6, 2012 at 22:39 | comment | added | Dylan Wilson | Hmmm- I'm having trouble finding a reference of the "stable" part of "stable model category" BUT this much I know: The derived category of any grothendieck abelian category is an "axiomatic stable homotopy category" in the sense of HPS... but the only way I see how to prove this at the moment is to go through $\infty$-categories... I imagine there's a more elementary way of doing this... | |
| Jun 6, 2012 at 22:24 | comment | added | Dylan Wilson | If you're willing to leave out the word "monoidal" until you get to the actual homotopy category, then this is actually pretty easy to do for just a ring, and it would be a good exercise. | |
| Jun 6, 2012 at 22:20 | comment | added | Dylan Wilson | @Jon: "Is there a stable homotopy structure on..." Of course! This applies in huge, huge generality. One can put a symmetric monoidal, stable model structure on the category of chain complexes of $\mathcal{O}$-modules on any ringed space. | |
| Jun 6, 2012 at 20:48 | comment | added | Jonathan Beardsley | @Fernando thanks! I've read Neeman's paper, and a bit by another fellow Luke Wolcott. This is a really interesting topic! Seems like for non-Noetherian rings stuff gets complicated quick :) | |
| Jun 6, 2012 at 20:29 | comment | added | Fernando Muro | @Jon Most triangulated categories come from a stable model category, but the derived category of a non-noetherian ring is still very much unknown. The first stamente is purely formal, while the second is sort of 'down-to-earth'. | |
| Jun 6, 2012 at 20:00 | comment | added | Jonathan Beardsley | I have also just discovered an extremely brief discussion of it at: math.uwo.ca/~schebolu/apping/UW/future-research.pdf | |
| Jun 6, 2012 at 19:57 | comment | added | Jonathan Beardsley | @Fernando Muro No in fact I don't! I'll look into it. I guess I was just wondering if there was some specific algebraic obstruction, i.e. is there a stable homotopy structure on the derived category of a non-Noetherian ring? | |
| Jun 6, 2012 at 19:50 | comment | added | Fernando Muro | You question is somewhat vage, but if I understand correctly what you mean, the answer is that not only there's no general theory, but most problems and open and all answers seem to be possible. Probably you know Neeman's paper entitled "Oddball Bous"eld classes". | |
| Jun 6, 2012 at 19:43 | history | asked | Jonathan Beardsley | CC BY-SA 3.0 |