Let $R$ a commutative Noetherian ring and $M$ a finitely generated $R$-module such that $\operatorname{pd}_R(M)=d < \infty$, where $\operatorname{pd}_R(M)$ denotes the projective dimension of $M$.
Can I prove that there exists a projective resolution of $M$
$0 \to P_d \to P_{d-1} \to \cdots \to P_1 \to P_0 \to M \to 0$ where each $P_i$ is finitely generated?
Best Answer
Yes; this follows from two key facts:
Fact 1 If $R$ is Noetherian and $M$ is finitely generated then $M$ has a projective resolution consisting of finitely generated modules.
Proof Idea. Construct a free resolution in the standard way, and use the fact that $R$ is Noetherian to verify that the kernels are finitely generated at each step.
Fact 2 If $M$ has projective dimension $d < \infty$, then for any projective resolution $$\dots \to F_d \xrightarrow{f_d} \dots \to F_0 \xrightarrow{f_0} M \to 0,$$ $\ker f_d$ is projective.
Proof Idea. Use the generalized version of Schanuel's lemma with the assumed projective resolution of length $d$ and the given projective resolution.
A complete proof of these facts can be found in https://www.math.purdue.edu/~iswanso/homologicalalgebra.pdf (Theorem 6.5 and Proposition 6.6).