Let $R$ be a commutative ring and $S$ be a flat $R$ algebra.
I want to prove that for any $S$ module $N$ and for any $R$ module $M$, there is an isomorphism $\mathrm{Ext}_{S}^{m}(S \otimes_{R} M,N)\cong \mathrm{Ext}_{R}^{m}(M,N)$.
If $m=0$,it is easy because $$ \mathrm{Hom}_S(S\otimes_RM,N)\cong \mathrm{Hom}_S\bigl(S,\mathrm{Hom}_R(M,N)\bigr)\cong \mathrm{Hom}_R(M,N) $$.
How to prove $m>0$?