Local Cohen-Macaulay ring over which every finitely generated module of finite injective dimension also has finite projective dimension

Question

Let $(R,\mathfrak m,k)$ be a local Cohen-Macaulay ring. If every finitely generated $R$-module that has finite injective dimension also has finite projective dimension, then is it true that $R$ is Gorenstein ?

My approach: Let $d=\dim R(=\operatorname {depth } R)$ . Let $\{x_1,...,x_d\}$ be a system of parameters, then it is also a regular sequence of maximal length. Let $I=(x_1,...,x_d)$. So $N:= \operatorname {Hom}_R (R/I, E_R(k))$ is a finitely generated module (even has finite length) of finite injective dimension (where $E_R(-)$ denotes injective hull as $R$-module) . So by assumption $N$ has finite projective dimension. We also notice that $N$ has depth $0$. So by Auslander-Buchsbaum formula, $pd_R (N)=$depth $(R)-$depth$(N)=$depth $(R)$. I'm not sure if this would be useful or not.

Please help.

Answer 1

Your argument is very useful. I follow your argument. Since $R/I$ has finite length, we know $$R/I\cong \mathrm{Hom}_R(\mathrm{Hom}_R(R/I,E_R(k)),E_R(k))=\mathrm{Hom}_R(N,E_R(k)).$$ You have showed that $\mathrm{pd}_RN<\infty$, so we know $\mathrm{id}_R(R/I)<\infty$.

Lemma 1. Let $(R,m,k)$ be a Noetherian local ring. If $M$ is a finitely generated $R$-module, then $\mathrm{id}_RM=\sup\{i\mid \mathrm{Ext}^i_R(k,M)\neq 0\}$.

By lemma 1, and by Nakayama we know:

Lemma 2. Let $(R,m,k)$ be a Noetherian local ring, $M$ is a finitely generated $R$-module, and $x\in m$ is a regular element on $M$. Then $\mathrm{id}_R(M)=\mathrm{id}_R(M/xM)$.

So we know $\mathrm{id}_R(R)=\mathrm{id}_R(R/I)<\infty$, since $I$ is generated by a regular sequence.

More generally, Foxby showed:

If $R$ is a Noetherian local ring, if there exists a finitely generated module with finite injective dimension and finite projective dimension, then $R$ is Gorenstein.

Answer 2

This has been proved that if there exists a finitely generated $R$-module of finite projective dimension and finite injective dimension, then $R$ is Gorenstein (see Corollary 4.4 of "Isomorphisms between complexes with applications to the homological theory of modules" , Foxby). So the answer is positive.