Lemma 106.16.2. Let $\mathcal{X}$ be an algebraic stack with affine diagonal. Let $B$ be a ring. Let $F : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \text{Mod}_ B$ be a right exact tensor functor which commutes with direct sums. Let $g : U \to \mathcal{X}$ be a morphism with $U = \mathop{\mathrm{Spec}}(A)$ affine. Then
$C = F(g_{\mathit{QCoh}, *}\mathcal{O}_ U)$ is a commutative $B$-algebra and
there is a ring map $A \to C$
such that $F \circ g_{\mathit{QCoh}, *} : \text{Mod}_ A \to \text{Mod}_ B$ sends $M$ to $M \otimes _ A C$ seen as $B$-module.
Proof.
We note that $g$ is quasi-compact and quasi-separated, see Morphisms of Stacks, Lemma 101.7.8. In Cohomology of Stacks, Proposition 103.11.1 we have constructed the functor $g_{\mathit{QCoh}, *} : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$. By Cohomology of Stacks, Remarks 103.11.3 and 103.10.6 we obtain a multiplication
\[ \mu : g_{\mathit{QCoh}, *}\mathcal{O}_ U \otimes _{\mathcal{O}_\mathcal {X}} g_{\mathit{QCoh}, *}\mathcal{O}_ U \longrightarrow g_{\mathit{QCoh}, *}\mathcal{O}_ U \]
which turns $g_{\mathit{QCoh}, *}\mathcal{O}_ U$ into a commutative $\mathcal{O}_\mathcal {X}$-algebra. Hence $C = F(g_{\mathit{QCoh}, *}\mathcal{O}_ U)$ is a commutative algebra object in $\text{Mod}_ B$, in other words, $C$ is a commutative $B$-algebra. Observe that we have a map $\kappa : A \to \text{End}_{\mathcal{O}_\mathcal {X}}(g_{\mathit{QCoh}, *}\mathcal{O}_ U)$ such that for any $a \in A$ the diagram
\[ \xymatrix{ g_{\mathit{QCoh}, *}\mathcal{O}_ U \otimes _{\mathcal{O}_\mathcal {X}} g_{\mathit{QCoh}, *}\mathcal{O}_ U \ar[d]_{\kappa (r) \otimes 1} \ar[rr]_-\mu & & g_{\mathit{QCoh}, *}\mathcal{O}_ U \ar[d]^{\kappa (r)} \\ g_{\mathit{QCoh}, *}\mathcal{O}_ U \otimes _{\mathcal{O}_\mathcal {X}} g_{\mathit{QCoh}, *}\mathcal{O}_ U \ar[rr]^-\mu & & g_{\mathit{QCoh}, *}\mathcal{O}_ U } \]
commutes. It follows that we get a map $\kappa ' = F(\kappa ) : A \to \text{End}_ B(C)$ such that $\kappa '(a)(c) c' = \kappa '(a)(cc')$ and of course this means that $a \mapsto \kappa '(a)(1)$ is a ring map $A \to C$.
The morphism $g : U \to \mathcal{X}$ is affine, see Morphisms of Stacks, Lemma 101.9.4. Hence $g_{\mathit{QCoh}, *}$ is exact and commutes with direct sums by Cohomology of Stacks, Lemma 103.13.4. Thus $F \circ g_{\mathit{QCoh}, *} : \text{Mod}_ A \to \text{Mod}_ B$ is a right exact functor which commutes with direct sums and which sends $A$ to $C$. By Functors and Morphisms, Lemma 56.3.1 we see that the functor $F \circ g_{\mathit{QCoh}, *}$ sends an $A$-module $M$ to $M \otimes _ A C$ viewed as a $B$-module.
$\square$
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