Cohomology of Pointed Finite Tensor Categories

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.

💡 Research Summary

The paper investigates the finite generation property of cohomology algebras for pointed finite tensor categories that are coradically graded over abelian groups. The authors work within the framework of tensor categories over an algebraically closed field of characteristic zero and define the cohomology of a finite tensor category C as Ext⁎_C(1, V) for any object V, with the unit object denoted by 1. A category is said to have finitely generated cohomology (FGC) if H⁎(C, 1) is a finitely generated algebra and each H⁎(C, V) is a finitely generated module over H⁎(C, 1). This notion captures the conjecture of Etingof and Ostrik that every finite tensor category should satisfy FGC.

The authors focus on coradically graded pointed finite tensor categories, meaning that the category is equivalent to the category of finite‑dimensional comodules over a coradically graded coquasi‑Hopf algebra M. By the Tannakian reconstruction theorem, such an M exists with coradical (kG, ω), where G is a finite abelian group and ω a normalized 3‑cocycle. The classification of these coquasi‑Hopf algebras (cited from works