Finiteness properties of local cohomology modules over differentiable admissible algebras

Using the theory of D-modules, we prove some finiteness properties for local cohomology modules with respect to a family of supports on the spectrum of a k-algebra R which satisfies certain conditions introduced by L. Núñez-Betancourt in [15]. These results generalize in a sense most of the finiteness properties established by G. Lyubeznik in [9]. We also introduce the class of quasi-holonomic D-modules which properly contains the class of holonomic D-modules and is closed by taking submodules, quotients, extensions, direct limits, localizations at multiplicative subsets of R and local cohomology.