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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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