The Stacks project

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

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 17.29.2. This proves existence. We omit the proof of uniqueness. $\square$