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So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of mathematics. Of course, the precursor to this was the work done by Devinatz, Hopkins, and Smith in classifying the thick subcategories of the stable hootopy category of (finite) spectra. They famously used this classification to prove the periodicity theorem, and I can see why it is helpful: the classification theorem reduces the periodicity theorem down to the (still nontrivial!) task of finding a single type $n$ complex with a periodic self-map. I'm sure similar uses have been found for the other classification theorems, but I am left to wonder, more generally:

What kinds of problems are made simpler with a classification theorem? What questions does it answer?

I'm looking for some general heuristics here. They should satisfy the following conditions:

  1. They should apply to theorems already proven by classification theorems; examples and references would be lovely here!

  2. These heuristics should come with some sense of why one would think to use the classification theorem in this way. For example, I can see how the classification theorem makes the periodicity theorem manageable to prove, but why would one think to use it in the first place?

  3. This last one is more of a throw away or a bonus, but it's worth a shot: If there are any areas of mathematics, or open problems that you think are begging for a classification theorem type application then please share! It would be a useful test of the proposed heuristics if they are able to predict the solution of a problem that has not been solved...

Basically I'm looking for some intuition here. My logic being: if we know more about what kinds of questions a classification can answer then we will know more about the information contained in a classification. This, in turn, may provide clues for how to compute or construct such a classification (which is, of course, the next step in Balmer's program).

(P.S. I've tagged the areas that I know of with classification theorems. If I'm forgetting some, do remind me in the comments Looks like there's a limit on tags :).)

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2 Answers

I'm not completely sure if this is the sort of thing you are after, but the telescope conjecture (conjecture isn't a great word as it is known to be false for some categories) springs to mind as something one can (sometimes) answer using a classification theorem. The telescope conjecture holds for a compactly generated triangulated category $\mathcal{T}$ if every smashing subcategory $\mathcal{S}$ of $\mathcal{T}$, i.e. a coproduct closed triangulated subcategory whose inclusion admits a coproduct preserving right adjoint, is generated by compact objects of $\mathcal{T}$. It is obviously reasonable to attack this via a classification and this is precisely the key ingredient in the proof that the derived category of a commutative noetherian ring satisfies the telescope conjecture.

Another example, just because it is cool, is this paper by Ingalls and Thomas. They prove a classification for certain subcategories of the representation category of a quiver $Q$ of Dynkin type in terms of noncrossing partitions associated to $Q$ (this is done in the abelian setting but it extends to a classification of localizing subcategories of the derived category). They use this, for example, to give a new proof that the noncrossing partitions associated to $Q$ form a lattice.

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Let me go through a possible answer to my own question- just do some thinking out loud here. It seems to me that a classification of thick subcategories of a (tensor-) triangulated category is a good thing to have when you have some, relatively generic, thick subcategory, $S$, at hand and you want to prove something about it. Then you can say, "Well, $S$ can only be one of a couple subcategories and I have them all listed here. It can't be any of these because __, and if it's any of these other ones, then I can easily prove my claim.

Here's an example that I thought of, but I'm not sure if this has been done. Suppose you have some functor $F: \mathcal{T} \rightarrow \mathcal{A}$ from a triangulated category to an abelian category. If, for some reason, it turned out that the kernel of this functor was a thick subcategory then we could apply the thick subcategory theorem to ask the question "When is $Fa = 0$?" In the case that the functor at hand satisfies $Fa = 0$ only when $a = 0$, then perhaps you could run through your classification and find a single object in each nonzero thick subcategory that maps to something nontrivial in $\mathcal{A}$. This would then prove that the kernel has to be whatever thing you had left.

Actually, what's good about this use of the classification theorem is that it can be used very easily to get partial results. Maybe the thick subcategories of your triangulated category come in multiple flavours and it is easy to show that one flavour is not the kernel of the given functor, but you don't know for the other flavours. Maybe that's all you need to prove something useful regarding the functor at hand.

I'm not sure how many functors from triangulated categories to abelian categories happen to satisfy the property that their kernel is a thick subcategory... On the other hand, any (triangulated) functor between triangulated categories does satisfy this property. So maybe you could apply this technique in the following way: You have a map between two objects $A \rightarrow B$ in some category (like noetherian schemes, or finite group schemes, or topological spectra). This should give rise to (hopefully triangulated) functors betweensome triangulated categories related to $A$ and $B$ (like derived category of perfect complexes on a scheme, or stable module category, etc.) I'm pretty sure that, at least in the usual cases, the original map will be trivial if and only if this induced map is. To find out if the induced map is trivial, we may pull the same trick as before and check to see if the thick subcategory corresponding to the kernel must be zero.

That seems like a high-powered way to check if a map between to objects is trivial! But maybe it's useful? If anyone has seen anything like what I've described then please post a reference! I would love to know if this application has been done before (I'm sure it has, it was the first thing that came to mind...)

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