ac.commutative algebra – on the relative conductor of spherical singularity and quotient of beliefs retort

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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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