Cohomology of Pointed Finite Tensor Categories
Bowen Li, Gongxiang Liu
TL;DR
The paper investigates whether the cohomology $H^{\bullet}(\mathcal{C},\mathbf{1})$ of a finite tensor category $\mathcal{C}$ is finitely generated (FGC), focusing on coradically graded pointed finite tensor categories over finite abelian groups. It classifies coradically graded coquasi-Hopf algebras into diagonal and non-diagonal types and proves FGC for both by combining de-equivariantization (reducing diagonal type to Hopf-algebra cases) with exact-sequence techniques that transfer FG properties to non-diagonal cases. The diagonal-type analysis leverages a bridge to diagonal Nichols algebras via an extended abelian group, while the non-diagonal case is handled through an exact sequence that relates it to a diagonal-type framework. Together, these results confirm the Etingof–Ostrik conjecture for this broad class and provide a framework for transferring cohomological finite generation along de-equivariantizations and exact tensor-category sequences.
Abstract
We consider the finite generation property for cohomology algebra of pointed finite tensor categories via de-equivariantization and exact sequence of finite tensor categories. As a result, we prove that all coradically graded pointed finite tensor categories over abelian groups have finitely generated cohomology.