$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated categories" in which Balmer proposes his notion of spectrum which nowadays is considered central in the understanding and classification of the homotopy categories which we want to study in the concrete mathematical practice (to name a few examples: the $G$-equivariant stable homotopy category for $G$ a compact Lie group, or the derived category of quasi-coherent sheaves on a scheme).
Since I am not familiar with this notion I wanted to ask here various questions about the underlying ideas of such concept.
(1) I noticed that all the examples proposed by Balmer in his paper deal with compact objects, in the sense that the proposed tensor triangulated categories (t.t. categories from now on) can be identified with the full-subcategories of compact objects in a larger t.t. category. And from what I remember every other example which I read in different sources does the same thing: we study the Balmer spectrum of compact objects in a larger t.t. category. Balmer does not explicitly state that this must be the case, indeed his definition does not require the involved objects to be compact a priori.
For this abstract machinery to work we only need the t.t. category to be essentially small. I could think that this is the problem: in general we cannot guarantee that the t.t. category we are interested in is essentially small so we restrict to the subcategory of its compact objects for this property to be more likely.
But I have other reasons to believe that this justification is not completely correct: if we indulge in the intuition suggested by the choice of words, we should think of the support of an object in our t.t. category as an higher categorical analogue of the usual support of a function. Fixing the domain of our functions to be compact spaces ensures that the support will also be compact. So if we consider also non-compact objects the support could be non "topologically small".
Thus I am inclined to believe for the complete t.t. categories either the Balmer spectrum is too big to be computed or its is not the correct notion we want to use to classify their tensor subcategories.
(2) Related to the previous question: if the proposed notion of Balmer spectrum should be applied only to categories of compact objects, what can we deduce about the whole category of possibly non-compact objects? Suppose we consider an essentially small t.t. category $\mathcal{T}$ and we manage to compute the Balmer spectrum of $\mathcal{T}^c$, can we deduce any information regarding the thick tensor ideals or localizing tensor ideals of $\mathcal{T}$?
Two classical examples of this are $D(R)$, the derived category of a commutative ring $R$, and $\SHC$, the stable homotopy category. For $D^{\perf}(R)$ this is homeomorphic to the usual Zariski spectrum $\Spec(R)$, while for $\SHC^\mathrm{c}$ we have the classification provided by the thick subcategory theorem from chromatic homotopy theory. But I have never seen a classification (even partial) of their thick tensor subcategories or thick localizing subcategories.
(3) What information does the Balmer spectrum encode? Balmer proves that there is a bijection between the Thomason subsets of this spectrum and the radical thick tensor ideals of the t.t. category. But other than this? At first I expected that if two t.t. categories had isomorphic spectrum then they would have a sufficiently compatible t.t. structure. Then I found the following interesting example: we have that the Balmer spectrum of the category of compact rational $S^1$-equivariant spectra is homeomorphic to $\Spec(\mathbb{Z})$. If $H \leq S^1$ is a closed subgroup then the kernel of $\phi^H$, the non-equivariant geometric $H$-fixed points, provides a Balmer prime. Then $\ker \phi^{S^1}$ corresponds to the generic point $(0)$, while $\ker \phi^{C_n}$ can be mapped to $(p_n)$ where we order the prime numbers $\{p_n : n \geq 1 \}$.
Therefore $S^1\text{-}\SHC^\mathrm{c}_{\mathbb{Q}}$ and $D^{\perf}(\mathbb{Z})$ have the same Balmer spectrum, but they are very different t.t. categories: for one, the latter has a compact generator given by the tensor unit, while this is not the case in the former category. I would have thought that the t.t. structure would have been more rigid with respect to the Balmer spectrum, but this seems not to be the case.
If you wanted a more precise question: if two t.t. categories have homeomorphic Balmer spectra, can we translate this to any information on the two categories? What if the homeomorphism is induced by a monoidal exact functor? Can we deduce it is fully faithful, essentially surjective or any other property?
I hope that my questions are not too vague or naïve.
Best Answer
I am not an expert in tt-geometry, but let me try to answer some of your questions.
(1) You are correct, the Balmer spectrum is typically not well-suited to study the "big" categories - this is because all definitions that appear only use "finitary" things : tensor products, cones/extensions, finite direct sums, retracts. This makes it, as defined, ill-suited for studying big categories where you also have interesting infinitary phenomena.
In a big category, you might be more interested in studying localizing ideals for instance, where you can take arbitrary (homotopy) colimits, but then you run into subtle issues about compact generation and telescope conjectures etc. (which are also studied !)
This is not a hopeless situation, though : a lot of work has been done (is probably being done) about finding suitable notions of support for "big" categories (see e.g. Balmer's paper Homological support of big objects in tensor-triangulated categories - this is far from the only one on the topic, see e.g. Big categories, big spectra by Balchin and Stevenson)
In fact, the place where the Balmer spectrum is somehow the best suited is when the monoidal structure interacts well with finiteness: namely in rigid situations (resp. rigidly compactly generated).
There was not a clear question here, so I hope this answers it.
(2) I think I've answered this partially in my answer to (1). Thick tensor ideals in the small world give rise to localizing ideals in the big world, but in general there is no way to go back, and even when there is the comparison is not perfect (a keyword here is telescope conjecture; but it's not the only thing, and the example of $SHC$ should be enlightening : say we look at a prime $p$, then the kernel of $K(n)\otimes -$ and $E_n \otimes -$ are very different, as witnessed by the difference between $L_{K(n)}$ and $L_n$, but their kernels agree in $SHC^c$). The Balchin-Stevenson paper I mentioned earlier has a section "Comparison maps". Probably other papers that study this kind of thing raise the same kind of question, so you might want to look at that literature (if someone more knowledgeable wants to edit my answer and add some references about this, they would be most welcome !).
The moral is somehow that "big" things are harder to classify.
(3) An abstract homeomorphism of spectra is unlikely to give you any information, except that the "large scale" structure of the two tt-categories is the same (but that would be tautological : one could define this large scale structure by the Balmer spectrum) . This is the same thing with ordinary commutative rings: an abstract homeomorphism of spectra won't tell you much.
You can say much more if the homeomorphism is induced by a tt-functor between them f course, and somehow the functoriality of the Balmer spectrum is key to Balmer's approach, and to computations (e.g. the computation of the spectrum of the equivariant stable homotopy category relies heavily on leveraging the various geometric fixed points functors that one has). If you think in terms of rings, a morphism of commutative rings $R\to S$ that induces a homeomorphism of spectra doesn't tell you that they are isomorphic : indeed, there is some nilpotent business happening here. But you can think of it as some going up/down theorem.
Balmer has a paper about this kind of question, called On the surjectivity of the map of spectra associated to a tensor-triangular functor. He proves there for instance that if the map of spectra is surjective (on closed points), then the original functor is conservative. He further completely characterizes surjectivity in terms of detection of nilpotence (which is related to the nilpotent problem I mentioned earlier for rings).
Certainly, more things can be said if the map is a homeomorphism, and again someone more aware of the literature on the topic could probably say more than I did (if anyone wants to edit and add some references, it would be great, as before).
As a general rule of thumb, a good place to test ideas/conjectures about this stuff is with commutative rings or more generally schemes : the spectrum of the perfect derived category of a (nice) scheme $X$ is exactly (the underlying space of) $X$, in a way compatible with morphisms of schemes. This is to some extent not the most interesting case, but it's a good way to check intuitions.