Let $M$ be a finitely generated module (over a local noetherian ring $(R,\mathfrak m))$ such that the projective dimension of $M$ is finite $(pd\ M=n<\infty)$. We know that
i) There is a free resolution $F_\bullet$ of $M$ in which the free $R$-modules $F_i$ are finitely generated,
ii) Every projective resolution of $M$ can be truncated to a projective resolution that have length $n$.
From these, can one show that there exists a projective resolution $$ 0\to P_n\to P_{n-1}\to\cdots\to P_0\to M $$ of $M$ for which $P_i$ are finitely generated for all $i\in\{1,\ldots, n\}$?
Thanks.
Let $$ \cdots\to F_m\to F_{m-1}\to\cdots\to F_0\to M\to0 $$ be a free resolution of $M$ with $F_i$ of finite rank for all $i\ge 0$. If one stops at step $n=pd\ M$ we have $$ 0\to K_{n-1}\to F_{n-1}\to\cdots\to F_0\to M\to0 $$ and $K_{n-1}$ must be projective. Furthermore, since $F_n\to K_{n-1}\to0$ we have that $K_{n-1}$ is finitely generated, so it is free of finite rank.