We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.133) and the corresponding section in the chapter on sheaves of modules (Modules, Section 17.29).
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 sending a differential operator to its action on global sections is bijective.
\[ \text{Diff}^ k_{X/S}(\mathcal{F}, \mathcal{G}) \to \text{Diff}^ k_{A/R}(\Gamma (X, \mathcal{F}), \Gamma (X, \mathcal{G})) \]
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. $\square$
Lemma 29.33.2. Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Let $\mathcal{F}$ and $\mathcal{F}'$ be quasi-coherent $\mathcal{O}_ X$-modules. Let $D : \mathcal{F} \to \mathcal{F}'$ be a differential operator of order $k$ on $X/S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Then there is a unique differential operator of order $k$ on $X \times _ S Y / Y$ such that $ D'(s \otimes t) = D(s) \otimes t $ for local sections $s$ of $\mathcal{F}$ and $t$ of $\mathcal{G}$.
\[ D' : \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \longrightarrow \text{pr}_1^*\mathcal{F}' \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \]
Proof. In case $X$, $Y$, and $S$ are affine, this follows, via Lemma 29.33.1, from the corresponding algebra result, see Algebra, Lemma 10.133.11. In general, one uses coverings by affines (for example as in Schemes, Lemma 26.17.4) to construct $D'$ globally. Details omitted. $\square$
Remark 29.33.3. Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Denote $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ the projections. In this remark, given an $\mathcal{O}_ X$-module $\mathcal{F}$ and an $\mathcal{O}_ Y$-module $\mathcal{G}$ let us set Denote $\mathcal{A}_{X/S}$ the additive category whose objects are quasi-coherent $\mathcal{O}_ X$-modules and whose morphisms are differential operators of finite order on $X/S$. Similarly for $\mathcal{A}_{Y/S}$ and $\mathcal{A}_{X \times _ S Y/S}$. The construction of Lemma 29.33.2 determines a functor which is bilinear on morphisms. If $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(B)$, and $S = \mathop{\mathrm{Spec}}(R)$, then via the identification of quasi-coherent sheaves with modules this functor is given by $(M, N) \mapsto M \otimes _ R N$ on objects and sends the morphism $(D, D') : (M, N) \to (M', N')$ to $D \otimes D' : M \otimes _ R N \to M' \otimes _ R N'$.
\[ \mathcal{F} \boxtimes \mathcal{G} = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \]
\[ \boxtimes : \mathcal{A}_{X/S} \times \mathcal{A}_{Y/S} \longrightarrow \mathcal{A}_{X \times _ S Y/S}, \quad (\mathcal{F}, \mathcal{G}) \longmapsto \mathcal{F} \boxtimes \mathcal{G} \]
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