The Stacks project

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

\[ \mathcal{F} \boxtimes \mathcal{G} = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \]

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

\[ \boxtimes : \mathcal{A}_{X/S} \times \mathcal{A}_{Y/S} \longrightarrow \mathcal{A}_{X \times _ S Y/S}, \quad (\mathcal{F}, \mathcal{G}) \longmapsto \mathcal{F} \boxtimes \mathcal{G} \]

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 $D \otimes D' : M \otimes _ R N \to M' \otimes _ R N'$.

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