The Stacks project

Remark 102.10.6. Let $\mathcal{X}$ be an algebraic stack. Given two quasi-coherent $\mathcal{O}_\mathcal {X}$-modules $\mathcal{F}$ and $\mathcal{G}$ the tensor product module $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ is quasi-coherent, see Sheaves on Stacks, Lemma 95.15.1 part (5). Similarly, given two locally quasi-coherent modules with the flat base change property, their tensor product has the same property, see Proposition 102.8.1. Thus the inclusion functors

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X}) \]

are functors of symmetric monoidal categories. What is more interesting is that the functor

\[ Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

is a functor of symmetric monoidal categories as well. Namely, given $\mathcal{F}$ and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we obtain

\[ \xymatrix{ Q(\mathcal{F}) \otimes _{\mathcal{O}_\mathcal {X}} Q(\mathcal{G}) \ar[rr] \ar[rd] & & \mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G} \\ & Q(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) \ar[ru] } \]

where the south-west arrow comes from the universal property of the north-west arrow (and the fact already mentioned that the object in the upper left corner is quasi-coherent). If we restrict this diagram to $U_{\acute{e}tale}$ for $U \to \mathcal{X}$ flat, then all three arrows become isomorphisms (see Lemmas 102.10.1 and 102.10.2 and Definition 102.9.1). Hence $Q(\mathcal{F}) \otimes _{\mathcal{O}_\mathcal {X}} Q(\mathcal{G}) \to Q(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G})$ is an isomorphism, see for example Lemma 102.4.2.

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