  We consider selfinjective Artin algebras whose cohomology groups are finitely
generated over a central ring of cohomology operators. For such an algebra, we
show that the representation dimension is strictly greater than the maximal
complexity occurring among its modules. This provides a unified approach to
computing lower bounds for the representation dimension of group algebras,
exterior algebras and Artin complete intersections. We also obtain new examples
of classes of algebras with arbitrarily large representation dimension.
