$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$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.