# A toy example of a tensor triangulated category?

I've been reading Paul Balmer's paper about constructing a "spectrum of prime ideals" on (essentially small) tensor triangulated category in order to then classify thick subcategories. This is all done to generalize work done in various fields throughout mathematics (e.g. Devinatz, Hopkins, and Smith's work in stable homotopy theory, and Pevtsova and Friedlander's work in finite group schemes). So the classic examples of tensor trianulated categories that Balmer talks about are the category of spectra of some space, the category of $G$-modules for some finite group scheme $G$, or the perfect derived category associated to a (topologically Noetherian) scheme (this is related to Thomalson's work reconstructing a scheme from the aforementioned category).

But I can't, for the life of me, think of more examples of tensor triangulated categories! (I'm new at all of this...) Can anyone give me a "toy" example of a tensor-triangulated category that is not an example of any of the ones I just listed? By "toy" example I mean that it should be relatively simple with an easy to understand structure. The purpose will be so that I can do Balmer's construction on the toy category to get a better understanding of what's going on.

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I think that the simplest example is $K(B)$ (the homotopy category of complexes; you can also consider $K^b(B)$) where $B$ is any tensor additive category. Certainly, this example is not independent from the ones you mentioned (yet note that a single additive $B$ could support more than one tensor structure).

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If you assume sufficient coherence and finiteness, I doubt that there are any other examples. If you take the subcategory generated by the unit and another object, endomorphisms of these probably form a Hopf algebroid in spectra and the subcategory they generate is perfect sheaves on the corresponding stack.

Here are two examples that don't quite fit your question, but which I think are instructive:

1. For a finite group, compare the category of perfect complexes, the bounded derived category of finitely generated modules, and their quotient, the stable category.

2. See what you can say the non-noetherian $k[x,x^{1/2},x^{1/3},\ldots]$.

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By "doubt there are any other examples" do you mean this precisely, as in- if a tensor triang. category satisfies [insert some conditions] then it arises as one of the already given examples. Or is this just a hunch? –  Dylan Wilson Jun 22 '10 at 18:45
Just a hunch, but I gave you a construction. By "sufficiently coherent" I meant a symmetric monoidal category enriched in spectra. Also, my construction assumes that the category is finitely generated; filtering a general category by finitely generated ones may not yield a good perspective. –  Ben Wieland Jun 22 '10 at 19:59
Modules over a monoid surely don't form sheaves on a stack of groupoids; you definitely don't get an antipode, but maybe you get everything else. You can still talk about sheaves on a stack of categories, but this may be less appealing. –  Ben Wieland Jun 22 '10 at 20:45

Not quite an answer.If you want to get a better understand what is going about reconstruction theorem, maybe you could take a look at this quesHow to unify various reconstruction theoremstion I asked.

I think the first step to understand these constructions is to take some really "trivial" example, such as $A-mod$,(I assume $A$ is commutative noetherian ring,but actually in abelian level, we do not need noethrian). Then you consider bounded derived category of $A-mod$, i.e. $D^b(A-mod)$. Take them as symmetric monoidal category. Then, calculate the spectrum of this triangulated category and then do the geometric realization.

Another example you can consider is $D^b(CohP^1)$. However, from my understanding, there is nothing much you can calculate. Because for the symmetric tensor triangulated category, the spectrum gets much simpler than non-symmetric case. It is direct imitation of prime spectrum of a commutative ring.

Moreover, I am not sure whether P.Balmer's reconstruction theorem works for non-noetherian case. (In abelian level, it does)

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Reconstructing a scheme using Balmer's machinery works for any quasi-compact quasi-separated scheme –  Greg Stevenson Jun 23 '10 at 0:19
@Greg: Thank you for pointing out! –  Shizhuo Zhang Jun 23 '10 at 7:35