Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras
Øyvind Solberg, Kent B. Vashaw, Sarah Witherspoon
TL;DR
The paper develops a noncommutative tensor triangular framework for bimodule and module stable categories, focusing on unipotent Hopf algebras and the stable left-right projective category. It establishes deep links between Balmer spectra, Hochschild cohomology, and Stevenson support, including homeomorphisms $ ext{Spc}( ext{E}) \cong ext{Proj}( ext{HH}^*(A))$ under plausible conjectures and shows how spectra map surjectively to classify thick subcategories and submodule structures. By constructing and analyzing dualities, functors between module and bimodule categories, and Rickard/idempotent machinery, the work connects abstract tensor-triangular geometry with concrete representation-theoretic invariants, especially for finite $p$-groups. The results yield Noetherian classifications of thick ideals under conjectural hypotheses and provide explicit spectrum computations in key unipotent Hopf-algebra examples, highlighting the role of the categorical center and Hochschild cohomology in noncommutative settings.
Abstract
We initiate a program aimed at classifying thick ideals, Balmer spectra, and submodule categories of various stable categories of bimodules and modules for finite dimensional selfinjective algebras, and at clarifying the relationship between the universal Balmer support and the Hochschild cohomology support. In this paper, we focus mostly on the case of a unipotent Hopf algebra $A$. The stable category $\underline{\mathsf{lrp}}(A^{\mathsf{env}})$ of $A$-bimodules that are projective as left and as right $A$-modules is a monoidal triangulated category under $\otimes_A$, and acts naturally on the stable category $\underline{\mathsf{mod}}(A)$ of $A$. We show in this case that the Balmer spectrum $\mathsf{Spc}(\mathcal{E})$ of the thick subcategory ${\mathcal E}$ of $\underline{\mathsf{lrp}}(A^{\mathsf{env}})$ generated by $A$ is homeomorphic to $\mathsf{Spc}(\underline{\mathsf{mod}}(A))$ and defines an embedding $\mathsf{Spc}(\underline{\mathsf{mod}}(A)) \to \mathsf{Spc}(\underline{\mathsf{mod}}(A^{\mathsf{env}}))$. Subject to a conjectural description of spectra of finite tensor categories, we show that the spectrum of ${\mathcal E}$ is homeomorphic to ${\mathsf{Proj}}$ of the Hochschild cohomology ring of $A$, and that the Hochschild support coincides with the universal Balmer support. We show that any subcategory ${\mathcal K}$ of $\underline{\mathsf{lrp}}(A^{\mathsf{env}})$ containing a thick generator admits a surjective continuous map from $\mathsf{Spc}(\underline{\mathsf{mod}}(A))$. As a consequence, under the aforementioned conjecture, this spectrum is Noetherian, classifies the thick ideals of ${\mathcal K}$, and classifies thick ${\mathcal K}$-submodule categories of $\underline{\mathsf{mod}}(A)$ via the Stevenson module-theoretic support. As examples, we present in detail the representations of finite $p$-groups.