On the Matlis duals of local cohomology modules

Abstract:

Let $(R,\mathfrak {m})$ be a Noetherian regular local ring of characteristic $p>0$ and let $I$ be a nonzero ideal of $R$. Let $D(-)= \operatorname {Hom}_R(-, E)$ be the Matlis dual functor, where $E = E_R(R/{\mathfrak {m}})$ is the injective hull of the residue field $R/{\mathfrak {m}}$. In this short note, we prove that if ${H}^i_I(R)\neq 0$, then $\operatorname {Supp}_R(D({H}^i_{I}(R)))=\operatorname {Spec}(R)$.