Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras

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.