The Stacks project

Lemma 29.33.1. Let $R \to A$ be a ring map. Denote $f : X \to S$ the corresponding morphism of affine schemes. Let $\mathcal{F}$ and $\mathcal{G}$ be $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is quasi-coherent then the map

\[ \text{Diff}^ k_{X/S}(\mathcal{F}, \mathcal{G}) \to \text{Diff}^ k_{A/R}(\Gamma (X, \mathcal{F}), \Gamma (X, \mathcal{G})) \]

sending a differential operator to its action on global sections is bijective.

Proof. Write $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$. Set $N = \Gamma (X, \mathcal{G})$. Let $D : M \to N$ be a differential operator of order $k$. We have to show there exists a unique differential operator $\mathcal{F} \to \mathcal{G}$ of order $k$ which gives rise to $D$ on global sections. Let $U = D(f) \subset X$ be a standard affine open. Then $\mathcal{F}(U) = M_ f$ is the localization. By Algebra, Lemma 10.133.10 the differential operator $D$ extends to a unique differential operator

\[ D_ f : \mathcal{F}(U) = \widetilde{M}(U) = M_ f \to N_ f = \widetilde{N}(U) \]

The uniqueness shows that these maps $D_ f$ glue to give a map of sheaves $\widetilde{M} \to \widetilde{N}$ on the basis of all standard opens of $X$. Hence we get a unique map of sheaves $\widetilde{D} : \widetilde{M} \to \widetilde{N}$ agreeing with these maps by the material in Sheaves, Section 6.30. Since $\widetilde{D}$ is given by differential operators of order $k$ on the standard opens, we find that $\widetilde{D}$ is a differential operator of order $k$ (small detail omitted). Finally, we can post-compose with the canonical $\mathcal{O}_ X$-module map $c : \widetilde{N} \to \mathcal{G}$ (Schemes, Lemma 26.7.1) to get $c \circ \widetilde{D} : \mathcal{F} \to \mathcal{G}$ which is a differential operator of order $k$ by Modules, Lemma 17.28.2. This proves existence. We omit the proof of uniqueness. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G44. Beware of the difference between the letter 'O' and the digit '0'.