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ac.commutative algebra – on the relative conductor of spherical singularity and quotient of beliefs
Ok, so it has been 7 years, however I assassinate maintain one thing recent so as to add to the solutions by Karl and Sándor.
All your questions are about whether or not some modules/beliefs are reflexive. That is as a result of for a fractional mannequin $I$ (together with any birational extension of $R$ or it is conductor), you’ll be able to determine $R:I$ with $I^*:=Hom_R(I,R)$. Then Question 2 asks if $R’$ is reflexive, Question 3 asks if $I$ is reflexive, and Question 4 asks if $I^*$ is reflexive.
(a diminutive carp: in Question 3) you wrote that the $R’= R:I$ is a subring, however I’m not optimistic, it is a ring. Perhaps you meant $R’=I:I$?)
Anyhow, as Karl’s retort indicated, if $R$ is Gorenstein then any torsion-free module is reflexive. So if $R$ is Gorenstein, as an example if it’s a planar spherical, the solutions are sure to all questions.
When $R$ will not be Gorenstein, surprisingly reflexive modules should not very well-understood. Recently I wrote a paper on this theme with a pair of collaborators, you could find it right here: https://arxiv.org/abs/2101.02641 and the references there.
You can make use of the outcomes there to win a few of the downhearted examples provided by Sándor. For occasion if $R=ok[[t^3,t^6,t^7…]]=ok[[t^3,t^7,t^8]]=ok[[x,y,z]]/P$, then the conductor is $c=(x^6,x^7,..)= (x^2,y,z)$. Theorem 3.5 of the above paper says that any reflexive mannequin would breathe isomorphic to an mannequin containing the conductor, so there are solely two succesful decisions, $c$ and the maximal mannequin $(x,y,z)$. Both are reflexive, and their endormorphism rings are the one reflexive birational extensions.
The retort to Question 4 is simply too sure in increased dimensions, you simply necessity $R$ to breathe $S_1$ and generically Gorenstein, graze Lemma 2.5 of the equivalent paper.
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