The Stacks project

[Lemma 1.8, Peskine-Szpiro]

Lemma 10.102.8 (Acyclicity lemma). Let $R$ be a local Noetherian ring. Let $0 \to M_ e \to M_{e-1} \to \ldots \to M_0$ be a complex of finite $R$-modules. Assume $\text{depth}(M_ i) \geq i$. Let $i$ be the largest index such that the complex is not exact at $M_ i$. If $i > 0$ then $\mathop{\mathrm{Ker}}(M_ i \to M_{i-1})/\mathop{\mathrm{Im}}(M_{i + 1} \to M_ i)$ has depth $\geq 1$.

Proof. Let $H = \mathop{\mathrm{Ker}}(M_ i \to M_{i-1})/\mathop{\mathrm{Im}}(M_{i + 1} \to M_ i)$ be the cohomology group in question. We may break the complex into short exact sequences $0 \to M_ e \to M_{e-1} \to K_{e-2} \to 0$, $0 \to K_ j \to M_ j \to K_{j-1} \to 0$, for $i + 2 \leq j \leq e-2 $, $0 \to K_{i + 1} \to M_{i + 1} \to B_ i \to 0$, $0 \to K_ i \to M_ i \to M_{i-1}$, and $0 \to B_ i \to K_ i \to H \to 0$. We proceed up through these complexes to prove the statements about depths, repeatedly using Lemma 10.72.6. First of all, since $\text{depth}(M_ e) \geq e$, and $\text{depth}(M_{e-1}) \geq e-1$ we deduce that $\text{depth}(K_{e-2}) \geq e - 1$. At this point the sequences $0 \to K_ j \to M_ j \to K_{j-1} \to 0$ for $i + 2 \leq j \leq e-2 $ imply similarly that $\text{depth}(K_{j-1}) \geq j$ for $i + 2 \leq j \leq e-2$. The sequence $0 \to K_{i + 1} \to M_{i + 1} \to B_ i \to 0$ then shows that $\text{depth}(B_ i) \geq i + 1$. The sequence $0 \to K_ i \to M_ i \to M_{i-1}$ shows that $\text{depth}(K_ i) \geq 1$ since $M_ i$ has depth $\geq i \geq 1$ by assumption. The sequence $0 \to B_ i \to K_ i \to H \to 0$ then implies the result. $\square$

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