Matlis duals of local cohomology modules and their endomorphism rings

Let ${(R, mathfrak{m})}Let (R, mathfrakm){(R, mathfrak{m})} denote a local ring. Let I ì R{I subset R} be an ideal with c =  grade I. Let D(·) denote the Matlis duality functor. In recent research there is an interest in the structure of the local cohomology module H^c_I : = H^c_I(R){H^c_I := H^c_I(R)}, in particular in the endomorphism ring of D(H^c_I){D(H^c_I)}. Let E _R (k) be the injective hull of the residue field R/mathfrakm{R/mathfrak{m}}. By investigating the natural map H^c_I ?D(H^c_I) ? E_R(k){H^c_I otimes D(H^c_I) to E_R(k)} we are able to prove that the endomorphism rings of D(H^c_I){D(H^c_I)} and of H^c_I{H^c_I} are naturally isomorphic. This natural homomorphism is related to a quasi-isomorphism of a certain complex. As applications we show results when the endomorphism ring of D(H^c_I){D(H^c_I)} is naturally isomorphic to R generalizing results known under the additional assumption of Hⁱ_I(R) = 0{H^i_I(R) = 0} for i 1 c{i not= c}.