Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let ${\sigma }_F\left(T\right)=\sigma \left(T\right)\setminus {\sigma }_e\left(T\right)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $Reg\left({\sigma }_F\left(T\right)\right)$ of all smooth points of ${\sigma }_F\left(T\right)$, there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^p({(z-T)}^k,E)$ grow at least like the sequence ${\left(k^d\right)}_{k\ge 1}$ with d = dim M.