Lemma 47.9.4. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet$ is an isomorphism for some $f \in I$, i.e., $K^\bullet$ is a complex of $A_ f$-modules. Then $R\Gamma _ Z(K^\bullet ) = 0$.

Proof. Namely, in this case the cohomology modules of $R\Gamma _ Z(K^\bullet )$ are both $f$-power torsion and $f$ acts by automorphisms. Hence the cohomology modules are zero and hence the object is zero. $\square$

There are also:

• 2 comment(s) on Section 47.9: Local cohomology

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).