$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness criteria and Ext vanishing. A nice example is a vast literature around the Auslander-Reiten conjecture (ARC) (in the local case):

(ARC) For a Noetherian local ring $R$ and a finitely generated $R$-module $M$ such that $\Ext^i_R(M,M\oplus R)=0$ for all $i>0$, then $M$ is free.

Let $R$ be a Noetherian local ring and $M$ be finitely generated $R$-modules. The Auslander transpose of $M$ is defined as $$\Tr M:=\coker(F^*\rightarrow G^*)$$ where $G\rightarrow F\rightarrow M\rightarrow0$ is a presentation of $M$ and $\_^*=\Hom_R(\_,R)$. An interplay between freeness and Tor vanishing is the Yoshino freeness criterion: if $\Tor_1^R(Tr M,M)=0$, then $M$ is free.

I wonder if there is also an interplay between freeness criteria and vanishing of $\Ext_R^i(\Tr M,N)$ for some $R$-module $N$. Or even any sufficient condition for having $\Ext^i_R(\Tr M,N)=0$.