Let $R$ be a complete local domain of dimension $1$ with canonical module $\omega_R$. If $\text{Ext}^1_R(\omega_R, R)=0$, then does it follow that $R$ is Gorenstein ? Is it true at least for some more restrictive classes of rings (without assuming more vanishing)?
I know I will be done if I can show $\omega_R$ has finite projective dimension (i.e. free), or $R$ has finite injective dimension, but I am unable to show either. My thoughts so far:
Let $\text{Tr}(-)$ denote the Auslander transpose. The kernel of the natural map $\text{Tr}(\omega)\to \left(\text{Tr}(\omega) \right)^{**}$ is $\text{Ext}^1_R(\text{Tr Tr }\omega_R, R)\cong \text{Ext}^1_R(\omega_R, R)=0,$ so the natural evaluation map $\text{Tr}(\omega)\to \left(\text{Tr}(\omega) \right)^{**}$ is injective. There are also five term long exact sequences (https://arxiv.org/abs/1405.5188, See 4.1) from which I am able to conclude $\text{Ext}^1_R(\omega,-)\cong \text{Tor}_1^R(\text{Tr }\Omega \omega, -)$ and $\text{Tor}_1^R(\omega,-)\cong \text{Ext}^1_R(\text{Tr }\Omega \omega, -)$.
But apart from this, I am unable to conclude anything else.
Please help.