Lemma 29.32.4. Let $R \to A$ be a ring map. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules on $X = \mathop{\mathrm{Spec}}(A)$. Set $S = \mathop{\mathrm{Spec}}(R)$. The rule which associates to an $S$-derivation on $\mathcal{F}$ its action on global sections defines a bijection between the set of $S$-derivations of $\mathcal{F}$ and the set of $R$-derivations on $M = \Gamma (X, \mathcal{F})$.

**Proof.**
Let $D : A \to M$ be an $R$-derivation. We have to show there exists a unique $S$-derivation on $\mathcal{F}$ which gives rise to $D$ on global sections. Let $U = D(f) \subset X$ be a standard affine open. Any element of $\Gamma (U, \mathcal{O}_ X)$ is of the form $a/f^ n$ for some $a \in A$ and $n \geq 0$. By the Leibniz rule we have

in $\Gamma (U, \mathcal{F})$. Since $f$ acts invertibly on $\Gamma (U, \mathcal{F})$ this completely determines the value of $D(a/f^ n) \in \Gamma (U, \mathcal{F})$. This proves uniqueness. Existence follows by simply defining

and proving this has all the desired properties (on the basis of standard opens of $X$). Details omitted. $\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: