Lemma 10.104.9. Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$. Let $M$ be a finite $R$-module of depth $e$. There exists an exact complex

with each $F_ i$ finite free and $K$ maximal Cohen-Macaulay.

Lemma 10.104.9. Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$. Let $M$ be a finite $R$-module of depth $e$. There exists an exact complex

\[ 0 \to K \to F_{d-e-1} \to \ldots \to F_0 \to M \to 0 \]

with each $F_ i$ finite free and $K$ maximal Cohen-Macaulay.

**Proof.**
Immediate from the definition and Lemma 10.104.8.
$\square$

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