# if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is “Gorenstein”

Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete intersection or Regular?

thank you.

• Could you be more precise (I suppose you do not allow to add conditions like $M$ free). – Vinteuil Feb 26 '15 at 12:20
• i mean nontrivial assumptions. – user 1 Feb 26 '15 at 12:22
• 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. – Vinteuil Feb 27 '15 at 8:43
• @Vinteuil your answer is good but i can not read french. if the proof is short can you please write it here (in english)? – user 1 Feb 27 '15 at 8:46

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.