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As Sandor pointed out, a necessary condition is that the prime ideal $P$ is a complete intersection. Here is a proof that it is also sufficient. It will suffice to prove the following: Claim: Let $(R …

It is hard to give a useful answer. I suspect whatever you want/need would be more specific. In particular, details on how such $R$ arises in your research would make it easier to say something more c …

Ok, so it has been 7 years, but I do have something new to add to the answers by Karl and Sándor. All your questions are about whether some modules/ideals are reflexive. That is because for a fraction …

It should be noted that the answer is yes if $R$ is normal and $M$ is torsion-free. That is because of the: Fact: if a map $f:A \to B$ of reflexive modules is locally an isomorphism in codimension on …

The authors reduce to the case of $R$ complete with infinite residue field and use them implicitly at a couple of places in the proof. This is a fairly standard practice. For instance, to assert that …

We use the fact that in an Artinian Gorenstein ring, any ideal contains the socle. The assumption tells us that the socle of $A$ is $\mathfrak m^2$, which is principal. Let $I\neq (0)$ be a non-maxima …