The Stacks project

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

\[ D' : \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \longrightarrow \text{pr}_1^*\mathcal{F}' \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G} \]

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 $\mathcal{G}$.

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. $\square$

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