Matlis duals of local cohomology modules and their endomorphism rings

Abstract

Let \({(R, \mathfrak{m})}\) denote a local ring. Let \({I \subset R}\) 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 \({H^c_I := H^c_I(R)}\), in particular in the endomorphism ring of \({D(H^c_I)}\). Let ER(k) be the injective hull of the residue field \({R/\mathfrak{m}}\). By investigating the natural map \({H^c_I \otimes D(H^c_I) \to E_R(k)}\) we are able to prove that the endomorphism rings of \({D(H^c_I)}\) and of \({H^c_I}\) 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 \({D(H^c_I)}\) is naturally isomorphic to R generalizing results known under the additional assumption of \({H^i_I(R) = 0}\) for \({i \not= c}\).