1

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?

Community
  • 1
Manos
  • 25,833
  • As Youngsu said the answer is NO. But note that "the $ d-th$ syzygy of $M$" in problem is $0$ or $MCM$ even when free resolution is infinite . – user 1 Mar 09 '14 at 11:34

1 Answers1

1

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.

Community
  • 1
Youngsu
  • 3,132