The Stacks project

Exercise 111.8.1. Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $M$ be an $A$-module. Let $x_1, \ldots , x_ n \in M$. Assume that

  1. $M/IM$ is generated by $x_1, \ldots , x_ n$,

  2. $M$ is a finite $A$-module,

  3. $I$ is contained in every maximal ideal of $A$.

Show that $x_1, \ldots , x_ n$ generate $M$. (Suggested solution: Reduce to a localization at a maximal ideal of $A$ using Exercise 111.7.2 and exactness of localization. Then reduce to the statement of Nakayama's lemma in the lectures by looking at the quotient of $M$ by the submodule generated by $x_1, \ldots , x_ n$.)

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Your email address will not be published.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 076D. Beware of the difference between the letter 'O' and the digit '0'.


View Exercise 111.8.1 as pdf