2013/11/07 by Waqas Mahmood, Mahmood, Waqas
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC
paper · pdf · doi:10.48550/arxiv.1311.1573
9 pages,to be appeared in International Electronic Journal of Algebra
arxiv created 2013/11/07 · arxiv updated 2013/11/08
For a Noetherian local ring (R,\mathfrak m) with \mathfrak p∈ \Spec(R) we denote ER(R/\mathfrak p) by the R-injective hull of R/\mathfrak p. We will show that it has an R^\mathfrak p-module structure and there is an isomorphism ER(R/\mathfrak p)≅ E_R^\mathfrak p(R^\mathfrak p/\mathfrak pR^\mathfrak p) where R^\mathfrak p stands for the \mathfrak p-adic completion of R. Moreover for a complete Cohen-Macaulay ring R the module D(ER(R/\mathfrak p)) is isomorphic to R_\mathfrakp provided that dim(R/\mathfrak p)=1 and D(⋅) denotes the Matlis dual functor \HomR(⋅, ER(R/\mathfrak m)). Here R_\mathfrakp denotes the completion of R_\mathfrak p with respect to the maximal ideal \mathfrak pR_\mathfrak p. These results extend those of Matlis (see \citem) shown in the case of the maximal ideal \mathfrak m.