The Stacks project

Lemma 15.31.1. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be an Koszul-regular sequence. Then the extended alternating Čech complex $R \to \bigoplus \nolimits _{i_0} R_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r}$ from Section 15.29 only has cohomology in degree $r$.

Proof. By Lemma 15.30.4 and induction the sequence $f_1, \ldots , f_{r - 1}, f_ r^ n$ is Koszul regular for all $n \geq 1$. By Lemma 15.28.4 any permutation of a Koszul regular sequence is a Koszul regular sequence. Hence we see that we may replace any (or all) $f_ i$ by its $n$th power and still have a Koszul regular sequence. Thus $K_\bullet (R, f_1^ n, \ldots , f_ r^ n)$ has nonzero cohomology only in homological degree $0$. This implies what we want by Lemma 15.29.6. $\square$

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