Let $R$ be a commutative Noetherian ring. For a finitely generated non-zero $R$-module $M$, one defines $$\text{grade}_R(M):=\inf \{j: \text{Ext}^j_R(M,R)\ne 0\} $$
My question is: Is it true that for any finitely generated non-zero $R$-module $M$, there exists a finitely generated $R$-module $F_M$ (depending on $M$) such that there exists a surjection $F_M \to M \to 0$, where $\text{Ext}^j_R(F_M,R)=0, \forall j\ne \text{grade}_R(M)$, and $F_M \cong \text{Ext}^{\text{grade}_R(M)}_R(F_M,R)$ ?