The Stacks project

Lemma 47.8.3. 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 $R\Gamma _ I(K^\bullet ) = 0$.

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


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