The Stacks project

Lemma 10.99.5. Let $R \to S$ be a local homomorphism of local Noetherian rings. Let $\mathfrak m$ be the maximal ideal of $R$. Let $0 \to F_ e \to F_{e-1} \to \ldots \to F_0$ be a finite complex of finite $S$-modules. Assume that each $F_ i$ is $R$-flat, and that the complex $0 \to F_ e/\mathfrak m F_ e \to F_{e-1}/\mathfrak m F_{e-1} \to \ldots \to F_0 / \mathfrak m F_0$ is exact. Then $0 \to F_ e \to F_{e-1} \to \ldots \to F_0$ is exact, and moreover the module $\mathop{\mathrm{Coker}}(F_1 \to F_0)$ is $R$-flat.

Proof. By induction on $e$. If $e = 1$, then this is exactly Lemma 10.99.1. If $e > 1$, we see by Lemma 10.99.1 that $F_ e \to F_{e-1}$ is injective and that $C = \mathop{\mathrm{Coker}}(F_ e \to F_{e-1})$ is a finite $S$-module flat over $R$. Hence we can apply the induction hypothesis to the complex $0 \to C \to F_{e-2} \to \ldots \to F_0$. We deduce that $C \to F_{e-2}$ is injective and the exactness of the complex follows, as well as the flatness of the cokernel of $F_1 \to F_0$. $\square$

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