Lemma 51.19.1. Let A be a ring. Let I \subset A be a finitely generated ideal. Set Z = V(I). For each derivation \theta : A \to A there exists a canonical additive operator D on the local cohomology modules H^ i_ Z(A) satisfying the Leibniz rule with respect to \theta .
Proof. Let f_1, \ldots , f_ r be elements generating I. Recall that R\Gamma _ Z(A) is computed by the complex
A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}
See Dualizing Complexes, Lemma 47.9.1. Since \theta extends uniquely to an additive operator on any localization of A satisfying the Leibniz rule with respect to \theta , the lemma is clear. \square
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