The complexity of crossed products

Let H be a finite-dimensional Hopf algebra, let A be a finite-dimensional algebra measured by H, and let A #σH be a crossed product. In this paper, we first show that if H is semisimple as well as its dual H*, then the complexity of A #σH is equal to that of A. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra H is equal to the complexity of the trivial module Hk. As an application, we prove that the complexity of Sweedler’s 4-dimensional Hopf algebra H4 is equal to 1.