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

TL;DR

This paper develops a noncommutative tensor triangular geometry framework for classifying thick ideals and spectra in stable categories of bimodules over unipotent Hopf algebras, linking Hochschild cohomology and support theories.

Contribution

It introduces a novel approach to classify thick ideals and spectra in stable bimodule categories for unipotent Hopf algebras, connecting Balmer spectra with Hochschild cohomology support.

Findings

The Balmer spectrum of certain subcategories is homeomorphic to the spectrum of Hochschild cohomology.

The spectrum of the subcategory generated by the algebra is homeomorphic to a projective scheme.

Under a conjecture, the spectrum is Noetherian and classifies thick ideals and subcategories.

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.