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).

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 $SHC^c$ 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 $S^1\text{-}SHC^c_{\mathbb{Q}}$ 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.

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

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.

But I have other reasons to believe that this justification is completely wrong: from what I remember $D(R)$, the derived category of a commutative ring $R$, and $SHC$, the stable homotopy category, are essentially small but we usually compute the Balmer spectrum of their compact objects. For $D^{perf}(R)$ this is homeomorphic to the usual Zariski spectrum $Spec(R)$, while for $SHC^c$ we have the classification provided by the thick subcategory theorem from chromatic homotopy theory.

(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 $\mathcal{T}$?

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".

(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 $SHC^c$ 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.