An application of generalized Matlis duality for quasi-\F-modules to the Artinianness of local cohomology modules

We use a result of Hellus about generalized local duality to describe some generalized Matlis duals for certain quasi-\F-modules. Furthermore, we apply this description to obtain examples for non-artinian local cohomology modules by the theory of \F-modules. In particular, we get a new view on Hartshorne’s counterexample for a conjecture by Grothendieck about the finiteness of $Ho{m}_R(R\mathrm{/}I,H_I^i(R))$ for a noetherian local Ring $R$ and an ideal $I \subseteq R$.