Let $(R,\mathfrak{m},k,E)$ be a Noetherian local ring with $E$ an injective hull of $k$, let $M$ be a finitely generated $R$-module and $M^{\vee}=\mathrm{Hom}_{R}(M,E)$. Is there a non-flat $R$-algebra $S$ such that $R\to k\cong F\to S$ where $F$ is a field and $$S\otimes _{R}\mathrm{Tor}_{*}^{R}(M,M^{\vee}\oplus E)\cong \mathrm{Tor}_{*}^{R}(M,S\otimes_{R}(M^{\vee}\oplus E))\ ?$$
I have been studing how to use a flatness criterion of Bhatt, Iyengar and Ma in order to prove the generalized Auslander-Reiten conjecture which says: if $\mathrm{Ext}_{R}^{j>n}(M,M\oplus R)=0$, then $\mathrm{pd}(M)\leq n.$
The hypothesis that $S$ is flat guarantees that $$S\otimes _{R}\mathrm{Tor}_{*}^{R}(M,M^{\vee}\oplus E)\cong \mathrm{Tor}_{*}^{R}(M,S\otimes_{R}(M^{\vee}\oplus E))$$ but also implies that $R$ is regular, in that case, this problem seems to be very simple by the work of Auslander et al. Then, the other possibility is try to find a non-flat algebra $S$ such that $R\to k\cong F\to S$ and $$S\otimes _{R}\mathrm{Tor}_{*}^{R}(M,M^{\vee}\oplus E)\cong \mathrm{Tor}_{*}^{R}(M,S\otimes_{R}(M^{\vee}\oplus E)),$$ but I don't know if such a thing is possible.
