  In support variety theory, representations of a finite dimensional (Hopf)
algebra $A$ can be studied geometrically by associating any representation of
$A$ to an algebraic variety using the cohomology ring of $A$. An essential
assumption in this theory is the finite generation condition for the cohomology
ring of $A$ and that for the corresponding modules. In this paper, we introduce
various approaches to study the finite generation condition. First, for any
finite dimensional Hopf algebra $A$, we show that the finite generation
condition on $A$-modules can be replaced by a condition on any affine
commutative $A$-module algebra $R$ under the assumption that $R$ is integral
over its invariant subring $R^A$. Next, we use a spectral sequence argument to
show that a finite generation condition holds for certain filtered, smash and
crossed product algebras in positive characteristic if the related spectral
sequences collapse. Finally, if $A$ is defined over a number field over the
rationals, we construct another finite dimensional Hopf algebra $A'$ over a
finite field, where $A$ can be viewed as a deformation of $A'$, and prove that
if the finite generation condition holds for $A'$, then the same condition
holds for $A$.
