# Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria

$$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$$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 $$\Ext^i_R(\Tr M,N)=0$$.

I am not sure if this is the kind of thing you are interested in, but let me at least state the easiest to prove criteria that I know for free-ness of $$M$$ in terms of vanishing of certain $$\text{Ext}_R^i(\text{Tr}M, -)$$ .
Proposition: If $$M$$ is a finitely generated module over a Noetherian local ring $$R$$ such that $$M^*\ne 0$$, and $$\text{Ext}_R^{1,2}(\text{Tr} M, M^*)=0$$, then $$M$$ is free. (See https://arxiv.org/abs/1805.04568 Lemma 4.12)
Proof: By definition, one have exact sequences $$0\to M^*\to F\to K \to 0$$ and $$0\to K \to G\to \text{ Tr} M\to 0$$ for some modules $$K,F,G$$, where $$F$$ and $$G$$ are free. Now $$\text{Ext}_R^{1}(K, M^*)\cong \text{Ext}_R^{2}(\text{Tr} M, M^*)=0$$, hence $$0\to M^*\to F\to K \to 0$$ splits, so $$M^*$$ is a free module. Since $$M^*$$ is non-zero, and now we know it is free, so $$\text{Ext}_R^{1,2}(\text{Tr} M, M^*)=0$$ yields $$\text{Ext}_R^{1,2}(\text{Tr} M, R)=0$$, hence $$M$$ is reflexive, so $$M\cong M^{**}$$ is free.
• I was just interested in a general interplay between the two subjects in my question so I still have no specific kind of situation to ask you about. Thank you. Further, Proposition 4.13 in your reference suggests sufficient conditions to have $Ext^{1,2}_R(Tr M,M^*)=0$. Nov 16, 2021 at 17:48