Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt)category. Balmer defined a functor from the category of ttcategories to the category of locally ringed spaces, called the Balmer spectra or ttspectra. 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 counterexample 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 ttcategories? Thank you in advance.
1 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 ttstructure $\boxtimes$ 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 ttcategories 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 $\otimes_{X}^{\mathbb{L}}$ ttstructure on $D^{b}(X)$.
You need some more control on either the variety or the sort of ttstructure you want to put on the triangulated category. For example disregard spaces like $X\amalg X$ or fix some conditions on the unit.

$\begingroup$ Thank you very much for your reference and suggestions. Is it straightforward to see that $Spc D^b(X \bigsqcup X, \boxtimes)$ is isomorphic to $X$ as ringed spaces? $\endgroup$– P. UsadaMay 18, 2023 at 2:14

1$\begingroup$ @P.Usada I think so. You have to keep in mind that the unit of this ttstructure is given by (Ox,0) and that as described, the spectrum is given by subcategories of the form Q+Db(X) with Q in the spec of Db(X), and so the open subsets of this space are too just of the form U+Db(X). Then you can calculate by hand the structure sheaf and check it must coincide with that of X. $\endgroup$– AT0May 18, 2023 at 7:05