A FEW COMMENTS ON MATLIS DUALITY

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For a Noetherian local ring (R, m) with p ∈ Spec(R), we denote the R-injective hull of R/p by ER(R/p). We show that it has an Rˆp -module structure, and there is an isomorphism ER(R/p) ∼= ERˆp (Rˆp/pRˆp ), where Rˆp stands for the p-adic completion of R. Moreover, for a complete Cohen-Macaulay ring R, the module D(ER(R/p)) is isomorphic to Rˆp provided that dim(R/p) = 1, where D(·) denotes the Matlis dual functor HomR(·, ER(R/m)). Here, Rˆp denotes the completion of Rp with respect to the maximal ideal pRp. These results extend those of Matlis (see [11]) shown in the case of the maximal ideal m.