I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete intersection or Regular?
thank you.
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$\begingroup$ Could you be more precise (I suppose you do not allow to add conditions like $M$ free). $\endgroup$– VinteuilCommented Feb 26, 2015 at 12:20
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$\begingroup$ i mean nontrivial assumptions. $\endgroup$– user 1Commented Feb 26, 2015 at 12:22
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$\begingroup$ I mean that I do not know in what kind of conditions are you thinking. For instance, the change from $M$ free to pd$M$ finite (Foxby) may seem a standard improvement, I do not know if you are interested in classes of modules as "test modules" (there is not a standard definition of test module, but see for instance arXiv:1405.5188), etc. $\endgroup$– VinteuilCommented Feb 27, 2015 at 8:43
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$\begingroup$ @Vinteuil your answer is good but i can not read french. if the proof is short can you please write it here (in english)? $\endgroup$– user 1Commented Feb 27, 2015 at 8:46
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$\begingroup$ If $R$ has dimension $1$ and $M$ is reflexive, then $R$ is Gorenstein. $\endgroup$– Dylan C. BeckCommented Aug 12, 2021 at 22:56
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2 Answers
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If you admit $M$ cyclic as additional assumtion, then $R$ is Gorenstein by a theorem in Peskine-Szpiro paper "Dimension projective finie et cohomologie locale", Theorem II.5.5.
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To complete Vinteuil's answer:
You can find translation of "Local cohomology and finite projective dimension", by C. Peskine and L. Szpiro, here. (Translator is Srikanth B. Iyengar).