In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-
modules for rings of invariants of finite groups in characteristic zero, and for strongly F-
regular finitely generated graded algebras with FFRT in prime characteristic. In each of
these cases, the ring itself, its localizations, and its local cohomology modules are
holonomic. We also show that holonomic D-modules, in this context, have finite length. We
obtain these results using a more general version of Bernstein filtrations.

In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic-
modules for rings of invariants of finite groups in characteristic zero, and for strongly-regular
finitely generated graded algebras with finite-representation type in prime characteristic. In
each of these cases, the ring itself, its localizations, and its local cohomology modules are
holonomic. We also show that holonomic-modules, in this context, have finite length, and we
prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these …