Attached primes and Matlis duals of local cohomology modules

Let $J$ be an ideal of a noetherian local ring $R$. We show new results on the set of attached primes ${\text{Att}}_R({\text{LCMo}}_J^l(R))$ of a local cohomology module ${\text{LCMo}}_J^l(R)$. To prove our results we establish and use new relations between the set ${\text{Att}}_R({\text{LCMo}}_J^l(R))$ of attached primes of a local cohomology module and the set ${\text{Ass}}_R(D({\text{LCMo}}_J^l(R)))$ of associated primes of the Matlis dual of the same local cohomology module.