$\textbf{Question:}$
Let $R\rightarrow S$ be a module-finite extension of complete local rings, with $S$ Regular. Prove that if $M$ is a reflexive $R-$module such that ${\rm Ext}^i_R(M^*,S) = 0$ for $i=1,…,n-2$, (where $n=\dim R$), then $M \in {\rm add}_R(S)$.
This is an exercise in the book ‘Cohen-Macaulay representations’. And I know the completeness in the condition implies that the KRS theorem holds for module category over $R$. Therefore, it is enough to prove that $M$ is a $R-$summand of a free $S-$module, or MCM $S-$module since $S$ is regular. But I don’t know how to deduce this.
The origin exercise does not describe the mean of letter ‘$n$’. I guess it represents $\dim R$, thus, I enclose the guess.