# Non-Noetherian Stable Homotopy

There seems to be quite a bit of theory developed to deal with "stable homotopy" in the sense of the derived category of a Noetherian ring, or just any situation where the endomorphism ring of the generator is Noetherian. While there are certainly obvious examples of non-Noetherian stable homotopy categories, are there any references giving a general theory? If not, what sort of things get in the way?

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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". –  Fernando Muro Jun 6 '12 at 19:50
I have also just discovered an extremely brief discussion of it at: math.uwo.ca/~schebolu/apping/UW/future-research.pdf –  Jon Beardsley Jun 6 '12 at 20:00
@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. –  Dylan Wilson Jun 6 '12 at 22:20
@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 –  David White Jun 7 '12 at 0:43
@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 –  Dylan Wilson Jun 7 '12 at 4:41

Alright, various things have been said in the comments, and I'd like to say something both coherent and correct since many of my comments above have been neither.

Here's what I understand about all of this.

First and foremost, unless I'm missing something big, it is absolutely the case that $D(R)$ is a stable homotopy category for any ring $R$. Here is a silly way-too-much-machinery reason why, with the benefit that it has references that I know of off the top of my head: The category of dg-modules over R (equiv. chain complexes) is Quillen equivalent to the category of HR-module spectra (http://homepages.math.uic.edu/~bshipley/zdga17.pdf). Thanks to, for example, May (pick a paper of his at random and it will probably contain this result), this has the structure of a symmetric monoidal stable model category, which turns its homotopy category into a stable homotopy category in the sense of HPS.

Second, if you're asking if people have a "general theory" for stable homotopy categories with a non-Noetherian endomorphism ring, the answer is absolutely yes, in many different guises. One can study symmetric monoidal stable model categories (of which there are many, most of which do not have a Noetherian endomorphism ring for generators). One can study symmetric monoidal stable $\infty$-categories which are basically the same. One can try to prove results similar to the nilpotence and classification theorem in these settings, this has been done for: (i) D(R) where R is any commutative ring [Thomason], (ii) D(R) where R is any epsilon-commutative, G-graded ring [Dell'Ambroglio, Stevenson], (iii) stmod(kG) where G is a finite group scheme [Friedlander-Pevtsova, Benson-Carlson-Rickard], (iv) D(X) where X is a quasi-compact, quasi-separated scheme [Thomason], and (v) $\mathcal{S}$ the category of finite spectra [Devinatz-Hopkins-Smith]. There are some others but I'm less familiar with them...

The thing that absolutely does not work, fails miserably actually, for non-Noetherian situations is an attempt to classify the localizing subcategories. Luke Wolcott knows a lot about how bad this can get (his very recent thesis was about it). I'm pretty sure Balmer has written some things about what one can say generally if you are in the Noetherian case. The point is that it's not even clear what a "general theory" would look like in the non-Noetherian case for stuff like localizing subcategories... again, Luke knows much more about this than me, so you should ask him. Fernando sums it up nicely in his original comment.

Finally, let me try to clear up two things I said in the comments (someone should correct me if I'm wrong):

1. The category of chain complexes on a Grothendieck abelian category can be given the structure of a stable model category in which the weak equivalences are the quasi-isos.

2. The category of chain complexes of $\mathcal{O}$-modules on any ringed space admits a symmetric monoidal model structure, which means that the unbounded derived category is at the very least a tensor-triangulated category (it's not immediately obvious that the tensor structure plays nice with the triangular structure, but it would be very strange to me if this wasn't true or obvious to someone else?)

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@Dylan Thanks! This is a great answer. I think that I should have been more clear in that the things I am really interested in are localizing subcategories and Bousfield lattices, which indeed, is stuff that Luke knows a lot about. Sometimes I forget that there is a lot more relevant information to categories besides their localizing subcategories ;-). I have read a little of Luke's work, where he shows that the situation can get really bad. I guess I just wanted to see if there was something general like HPS section 6 for non-Noetherian rings, but I think not. –  Jon Beardsley Jun 7 '12 at 16:27