Remark 103.11.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then there is a canonical commutative diagram
\[ \xymatrix{ f_{\mathit{QCoh}, *}\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_{\mathit{QCoh}, *}\mathcal{G} \ar[r] \ar[d] & f_*\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_*\mathcal{G} \ar[d]^ c \\ f_{\mathit{QCoh}, *}(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) \ar[r] & f_*(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) } \]
The vertical arrow $c$ on the right is the naive relative cup product (in degree $0$), see Cohomology on Sites, Section 21.33. The source and target of $c$ are in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Proposition 103.8.1. Applying $Q$ to $c$ we obtain the left vertical arrow as $Q$ commutes with tensor products, see Remark 103.10.6. This construction is functorial in $\mathcal{F}$ and $\mathcal{G}$.
Comments (0)