The Stacks project

Lemma 15.89.4. Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. For any $f_1, \ldots , f_ r \in R$ such that $V(f_1, \ldots , f_ r) = V(I)$

  1. the map of Koszul complexes $K(R, f_1, \ldots , f_ r) \to K(S, f_1, \ldots , f_ r)$ is a quasi-isomorphism, and

  2. The map of extended alternating Čech complexes

    \[ \xymatrix{ R \to \prod _{i_0} R_{f_{i_0}} \to \prod _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r} \ar[d] \\ S \to \prod _{i_0} S_{f_{i_0}} \to \prod _{i_0 < i_1} S_{f_{i_0}f_{i_1}} \to \ldots \to S_{f_1\ldots f_ r} } \]

    is a quasi-isomorphism.

Proof. In both cases we have a complex $K_\bullet $ of $R$ modules and we want to show that $K_\bullet \to K_\bullet \otimes _ R S$ is a quasi-isomorphism. By Lemma 15.89.2 and the flatness of $R \to S$ this will hold as soon as all homology groups of $K$ are $I$-power torsion. This is true for the Koszul complex by Lemma 15.28.6 and for the extended alternating Čech complex by Lemma 15.29.5. $\square$

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