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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.

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and you have tried....? – user38268 Jun 17 '12 at 9:17
up vote 4 down vote accepted

Denote by $\Omega_i(M)$ the $i$th syzygy of $M$.

Prove by induction on $i\ge 1$ that $\text{depth }\Omega_i(M)\ge\text{min}(i,\text{depth}(R))$. (Use short exact sequences and depth Lemma, i.e. Proposition 1.2.9 from B&H.)

In particular, $\text{depth }\Omega_d(M)\ge\text{min}(d,\text{depth}(R))=d$. On the other side, $\text{depth }\Omega_d(M)\le\dim\Omega_d(M)\le\dim R=d$, and this proves that $\Omega_d(M)$ is maximal CM.

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