Categorical dynamics on stable module categories

Abstract

Let $A$ be a finite connected graded cocommutative Hopf algebra over a field $k$. There is an endofunctor $\mathsf{tw}$ on the stable module category ${\mathrm{StMod}}_A$ of $A$ which twists the grading by $1$. We show the categorical entropy of $\mathsf{tw}$ is zero. We provide a lower bound for the categorical polynomial entropy of $\mathsf{tw}$ in terms of the Krull dimension of the cohomology of $A$, and an upper bound in terms of the existence of finite resolutions of $A$-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of the twist functor for examples of finite graded Hopf algebras over ${\mathbb{F}}_2$.

43 pages, comments welcome