The Stacks project

Theorem 105.16.8. Let $\mathcal{X}$ be a Noetherian algebraic stack with affine diagonal. Let $B$ be a Noetherian ring. Let $F : \text{Coh}(\mathcal{O}_\mathcal {X}) \to \text{Mod}^{fg}_ B$ be a right exact tensor functor. Then $F$ comes from a unique morphism $\mathop{\mathrm{Spec}}(B) \to \mathcal{X}$.

Proof. By Lemma 105.16.1 we can extend $F$ uniquely to a right exact tensor functor $F : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \text{Mod}_ B$ commuting with all direct susms. Then we can apply Lemma 105.16.7. $\square$


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