Abstract: In studies of finite-dimensional Hopf algebras one makes consistent use of a number of standard operations. The most fundamental of these operations include linear duality, (cocycle) deformation, and twisting by so-called Drinfeld twists. Many numerical invariants of Hopf algebras are known to be stable under these operations. However, one can see from examples, that the cohomology ring H*(A,k) for a finite-dimensional Hopf algebra A with trivial coefficients is not preserved under deformation or duality of A. In joint work with J. Plavnik we conjecture that, although the cohomology ring itself may vary, the Krull dimension of cohomology should be invariant under a general class of ``duality operations" which includes linear duality, deformation, and Drinfeld twisting. I this talk I will give the necessary definitions and examples, discuss the aforementioned conjecture, and provide some positive results obtained jointly with J. Plavnik.