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. $\square$

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).