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.