existence of a finite free resolution Problem 2.1.26 in Bruns and Herzog, CMR, reads as follows: "Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and $M$ a finite $R$-module. Deduce that the $d$-th syzygy of $M$ in an arbitrary finite free resolution is either $0$ or a maximal Cohen-Macaulay module."

Question: Are the hypotheses of the problem enough to guarantee that $M$ has a finite free resolution?

Remark: As a reminder, see also this related question what classes of modules admit finite free resolutions?

No, it is not enough. Take $M = R/\mathfrak m$, where $\mathfrak m$ is the maximal ideal. If $M$ has a finite free resolution, then $R$ is regular, and there are CM local rings which are not regular. For instance, take $R =\mathbb Z/(4)$ and $M =\mathbb Z/(2)$. 
Problem 2.1.26 asks a property of a syzygy module which in general is not free.
