Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt-)category. Balmer defined a functor from the category of tt-categories to the category of locally ringed spaces, called the Balmer spectra or tt-spectra. He showed that the functor is not essentially injective if we set the target to be the category of topological spaces, but I am wondering if there is any known counter-example in our setting. In particular, when the Bamler spectra $X$ of $(\mathcal T, \otimes)$ is a qcqs scheme (or a smooth proper variety over a field if that makes a difference), is it true that we have $(\mathcal T, \otimes) \simeq (Perf X, \otimes_{\mathcal O_X}^{\mathbb L})$ as tt-categories? Thank you in advance.
Balmer Spectra – Essential Injectivity
ag.algebraic-geometryderived-categoriestriangulated-categories
Related Solutions
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.
Best Answer
No this is not true in general. In Tensor Triangulated Categories in Algebraic Geometry, Prop 4.0.9, Sosna shows that if X is a connected noetherian scheme then it is possible to put a tt-structure $\boxtimes$ on the derived category of $X\amalg X$ such that $Spc(D^{b}(X\amalg X),\boxtimes)\cong X$, yet $D^{b}(X\amalg X)\not\simeq D^{b}(X)$ so there cannot be an equivalence as tt-categories between $(D^{b}(X),\otimes_{X}^{\mathbb{L}})$ and $D^{b}(X\amalg X,\boxtimes)$. This structure is just like a square zero extension for the $\otimes_{X}^{\mathbb{L}}$ tt-structure on $D^{b}(X)$.
You need some more control on either the variety or the sort of tt-structure you want to put on the triangulated category. For example disregard spaces like $X\amalg X$ or fix some conditions on the unit.