The Stacks project

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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