Question

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?

Answer 1

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?)