  We investigate Matlis duals of local cohomology modules and prove that, in
general, their zeroth Bass number with respect to the zero ideal is not finite.
We also prove that, somewhat surprisingly, if we apply local cohomology again
(i. e. to the Matlis dual of the local cohomology module), we get (under
certain hypotheses) either zero or $E$, an $R$-injective hull of the residue
field of the local ring $R$.
