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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GRR. Beware of the difference between the letter 'O' and the digit '0'.