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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). $\endgroup$ – Vinteuil Feb 26 '15 at 12:20
  • $\begingroup$ i mean nontrivial assumptions. $\endgroup$ – user 1 Feb 26 '15 at 12:22
  • $\begingroup$ 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. $\endgroup$ – Vinteuil Feb 27 '15 at 8:43
  • $\begingroup$ @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 1 Feb 27 '15 at 8:46
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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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  • $\begingroup$ thank you. do you have a link? (in english) $\endgroup$ – user 1 Feb 26 '15 at 12:27
  • $\begingroup$ No. You can find the original (in french) in numdam.org. $\endgroup$ – Vinteuil Feb 27 '15 at 8:44
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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).

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