The Stacks project

Exercise 111.4.1. Let $R$ be a ring. With notation as above.

  1. Give an example of a ring $R$ and an additive functor $F : \text{Mod}_ R \to \text{Mod}_ R$ which is not $R$-linear.

  2. Let $N$ be an $R$-module. Show that the functor $F(M) = M \otimes _ R N$ is $R$-linear, right exact, and commutes with direct sums,

  3. Conversely, show that any functor $F : \text{Mod}_ R \to \text{Mod}_ R$ which is $R$-linear, right exact, and commutes with direct sums is of the form $F(M) = M \otimes _ R N$ for some $R$-module $N$.

  4. Show that if in (3) we drop the assumption that $F$ commutes with direct sums, then the conclusion no longer holds.

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Your email address will not be published.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


View Exercise 111.4.1 as pdf