A toy example of a tensor triangulated category?

Question

I've been reading Paul Balmer's paper about constructing a "spectrum of prime ideals" on an (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). The classic examples of tensor triangulated categories that Balmer talks about are the category of spectra, 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 Thomason'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.

Answer 1

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).

Answer 2

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 $k[x,x^{1/2},x^{1/3},\ldots]$.

Answer 3

Take a finite dimensional Hopf algebra $H$, the category of $H$-modules is Frobenius (projectives=injectives and there is enough of both); e.g. take $H$ to be the group algebra of a finite group. So the stable category is triangulated (see Happel's book). In these examples, the usual tensor product of $H$-modules is well defined in the stable category.

This is explained in Khovanov's paper here. and developed further by You Qi. I recently upload a preprint considering co-Frobenius Hopf algebras (i.e. not necesarily finite dimensional Hopf algebras admitting a (left or right) integral $\Lambda\in H^*$, https://arxiv.org/abs/1904.10430) The category of comodules over a co-Frobenius Hopf algebra is Frobenius, the stable category is triangulated and the standard tensor product of comodules is well defined in the stable category.

Answer 4

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 question I asked: How to unify various reconstruction theorems.

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 noetherian). 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)