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3. Balmer spectrum/TTG

Questions related to tensor triangular geometry or the Balmer spectrum.


    1. Add remark

      Problem 3.1.

      [Vera Serganova] Compute the Balmer spectrum of the stable module categories of some pointed Hopf algebras of tame type in characteristic $0$ (e.g. $u_q(\mathfrak{sl}(1|2))$).


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          Problem 3.2.

          [Kent Varshaw] Let $\mathcal{C}$ be a symmetric FTC. Is it true that $\text{Spc}(\text{St}(\mathcal{C})) = \text{Proj}(\text{Ext}^*_{\mathcal{C}}(\mathbf{1}))$? (This is known for cocommutative Hopf algebras.) In the non-braided case, is it true that $\text{Spc}(\text{St}(\mathcal{C})) = \text{Proj}(\text{some particular subalgebra of Ext}^*_{\mathcal{C}}(\mathbf{1}))$?


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              Problem 3.3.

              [Julia Pevtsova] Denote the ’particular subalgebra of Ext’ from above question as $C^*$. For pointed Hopf algebras $H$ (set $\mathcal{C} = \text{Rep}(H)$) with abelian grouplikes in characteristic $0$, is it true that $C^* = H^*(\mathcal{C})$ modulo nilpotents? In other words, is $\text{Proj}(H^*(\mathcal{C})) = \text{Proj}(C^*)$?


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                  Problem 3.4.

                  [Pablo Ocal] Take $\mathbf{K}$ to be a tensor triangular category with finite Krull-Schmidt dimension. Then, is the Balmer spectrum is Noetherian? (See the work of Negron and Stevenson.)


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                      Problem 3.5.

                      [Julia Pevtsova] Let $\mathcal{C}$ be an FTC. Is $\text{Spc}(\text{Stmod}(\mathcal{C}))$ independent of the monoidal structure? (e.g. is it true for the quasi-Hopf modifications of small quantum groups $u_q^{\phi}(\mathcal{g})$?)


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                          Problem 3.6.

                          [Antoine Touze] What are the relationships between finite generation, Noetherianity, and topological Noetherianity? (Does this imply the tensor triangular property?)

                              Cite this as: AimPL: Finite tensor categories: their cohomology and geometry, available at http://aimpl.org/finitetensor.