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
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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.