A note on the Matlis dual of a certain injective hull

2013/06/14 by Schenzel, Peter
#13C11 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13D45 #Secondary: 13D45

paper · doi:10.48550/arxiv.1306.3311

Abstract

Let (R,\mathfrakm) denote a local ring with E = E_R(R/\mathfrakm) the injective hull of the residue field. Let \mathfrakp ∈ \Spec R denote a prime ideal with dim R/\mathfrakp = 1, and let E_R(R/\mathfrakp) be the injective hull of R/\mathfrakp. As the main result we prove that the Matlis dual \Hom_R(E_R(R/\mathfrakp), E) is isomorphic to R_\mathfrakp, the completion of R_\mathfrakp, if and only if R/\mathfrakp is complete. In the case of R a one dimensional domain there is a complete description of Q ⊗_R R in terms of the completion R.

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