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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: