# f.g. module $M$ over a complete local CM ring of dimension 1 such that $M, \text{Hom}_R(M,M), \text{Ext}^1_R(M,M)$ have finite injective dimension

Let $$(R,\mathfrak m)$$ be a local, $$\mathfrak m$$-adically complete, Cohen-Macaulay ring of dimension $$1$$. Assume that there exists a finitely generated $$R$$-module $$M$$ of depth $$0$$ such that $$M$$, $$\text{Hom}_R(M,M)$$ and $$\text{Ext}^1_R(M,M)$$ all have finite injective dimension.

Then, is it true that $$M$$ has finite projective dimension, i.e., is it true that $$R$$ is Gorenstein?

My thoughts: As $$R$$ is complete local and Cohen-Macaulay, $$R$$ admits a canonical module, say $$\omega$$. By MCM approximation, we get an exact sequence $$0\to X \to Y \to M \to 0$$, where $$Y$$ is MCM and $$X$$ has finite injective dimension. Note that as $$M$$ is not MCM, so $$X$$ is non-zero. By depth lemma, we get $$X$$ is also MCM. Hence $$X\cong \omega^{\oplus n}$$ for some $$n>0$$. As $$Y$$ also have finite injective dimension, so $$Y\cong \omega^{\oplus g}$$ for some $$g>0$$. So, we have an exact sequence $$0\to \omega^{\oplus n} \to \omega^{\oplus g} \to M\to 0$$. Applying $$\text{Hom}_R(\omega,-)$$, and remembering $$\text{Ext}^{>0}_R(\omega, \omega)=0$$ , we get an exact sequence $$0\to R^{\oplus n}\to R^{\oplus g}\to \text{Hom}_R(\omega, M)\to 0$$ Hence, $$\text{Hom}_R(\omega, M)$$ has projective dimension $$1$$, so has depth $$0$$. Also, applying $$\text{Hom}_R(-,M)$$ to $$0\to \omega^{\oplus n} \to \omega^{\oplus g} \to M\to 0$$ , and remembering $$\text{Ext}^{>0}_R(\omega, M)=0$$ (since $$\omega$$ is MCM and $$M$$ has finite injective dimension), we get an exact sequence $$0\to \text{Hom}_R(M,M)\to \text{Hom}_R(\omega, M)^{\oplus g}\to \text{Hom}_R(\omega, M)^{\oplus n}\to \text{Ext}^1_R(M, M)\to 0$$

As $$\text{Hom}_R(M,M)$$ and $$\text{Ext}^1_R(M, M)$$ also have finite injective dimension, so we also have exact sequences $$0\to \omega^{\oplus a} \to \omega^{\oplus b} \to \text{Hom}_R(M,M) \to 0$$ and

$$0\to \omega^{\oplus f} \to \omega^{\oplus h} \to \text{Ext}^1_R(M,M) \to 0$$ , where $$a,b,f,h$$ are non-negative integers. And now I am stuck, and do not know how to proceed further. Please help.