TITLE: Criterion for being reflexive via Ext
QUESTION [7 upvotes]: In this question it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a proof for this? If it is not correct, is there another criterion for being reflexive via some $\rm Ext$-goups vanishing?

REPLY [5 votes]: About the last question, I should point out that there is indeed a Ext-vanishing criterion for reflexivity. Namely, $M$ is reflexive iff $Ext^{1,2}(Tr(M),R)=0$ where $Tr(M)$ is the Auslander-Bridger transpose of $M$. For details see for instance Lemma 2.2 in this paper: https://arxiv.org/pdf/0809.1958v3.pdf