The Stacks project

Lemma 10.102.7. In Situation 10.102.1. Suppose $R$ is a local ring with maximal ideal $\mathfrak m$. Suppose that $0 \to R^{n_ e} \to R^{n_{e-1}} \to \ldots \to R^{n_0}$ is exact at $R^{n_ e}, \ldots , R^{n_1}$. Let $x \in \mathfrak m$ be a nonzerodivisor. The complex $0 \to (R/xR)^{n_ e} \to \ldots \to (R/xR)^{n_1}$ is exact at $(R/xR)^{n_ e}, \ldots , (R/xR)^{n_2}$.

Proof. Denote $F_\bullet $ the complex with terms $F_ i = R^{n_ i}$ and differential given by $\varphi _ i$. Then we have a short exact sequence of complexes

\[ 0 \to F_\bullet \xrightarrow {x} F_\bullet \to F_\bullet /xF_\bullet \to 0 \]

Applying the snake lemma we get a long exact sequence

\[ H_ i(F_\bullet ) \xrightarrow {x} H_ i(F_\bullet ) \to H_ i(F_\bullet /xF_\bullet ) \to H_{i - 1}(F_\bullet ) \xrightarrow {x} H_{i - 1}(F_\bullet ) \]

The lemma follows. $\square$

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