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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