The Stacks project

Lemma 10.121.1. Let $R$ be a semi-local Noetherian ring of dimension $1$. If $a, b \in R$ are nonzerodivisors then

\[ \text{length}_ R(R/(ab)) = \text{length}_ R(R/(a)) + \text{length}_ R(R/(b)) \]

and these lengths are finite.

Proof. We saw the finiteness in Lemma 10.119.11. Additivity holds since there is a short exact sequence $0 \to R/(a) \to R/(ab) \to R/(b) \to 0$ where the first map is given by multiplication by $b$. (Use length is additive, see Lemma 10.52.3.) $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 10.121: Orders of vanishing

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Your email address will not be published.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 02MC. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 10.121.1 as pdf