The Stacks project

Lemma 56.4.6. Let $A$ and $B$ be Noetherian rings. Let $F : \text{Mod}^{fg}_ A \to \text{Mod}^{fg}_ B$ be a functor. Then $F$ extends uniquely to a functor $F' : \text{Mod}_ A \to \text{Mod}_ B$ which commutes with filtered colimits. If $F$ is additive, then $F'$ is additive and commutes with arbitrary direct sums. If $F$ is exact, left exact, or right exact, so is $F'$.

Proof. See Lemmas 56.4.3 and 56.4.5. Also, use the finite $A$-modules are finitely presented $A$-modules, see Algebra, Lemma 10.31.4, and use that Noetherian rings are coherent, see Algebra, Lemma 10.90.5. $\square$

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Your email address will not be published.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 0GP9. Beware of the difference between the letter 'O' and the digit '0'.