The Stacks project

Lemma 36.6.9. Let $X$ be a scheme. Let $T \subset X$ be a closed subset which can locally be cut out by a Koszul regular sequence having $c$ elements. Then $\mathcal{H}^ i_ Z(\mathcal{F}) = 0$ for $i \not= c$ for every flat, quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. By the description of $R\mathcal{H}_ Z(\mathcal{F})$ given in Lemma 36.6.7 this boils down to the following algebra statement: given a ring $R$, a Koszul regular sequence $f_1, \ldots , f_ c \in R$, and a flat $R$-module $M$, the extended alternating Čech complex $M \to \bigoplus \nolimits _{i_0} M_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1 \ldots f_ c}$ from More on Algebra, Section 15.29 only has cohomology in degree $c$. By More on Algebra, Lemma 15.31.1 we obtain the desired vanishing for the extended alternating Čech complex of $R$. Since the complex for $M$ is obtained by tensoring this with the flat $R$-module $M$ (More on Algebra, Lemma 15.29.2) we conclude. $\square$


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