A FEW COMMENTS ON MATLIS DUALITY

Year 2014, Volume: 15 Issue: 15, 66 - 76, 01.06.2014

Waqas Mahmood

https://doi.org/10.24330/ieja.266238

https://izlik.org/JA28XK44ZM

Abstract

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.

Keywords

Matlis duality , injective hull , local cohomology , flat covers

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