The Stacks project

Remarks 110.49.2. Here are some trivial remarks.

  1. On a Noetherian integral scheme $X$ the sheaf ${\mathcal K}_ X$ is constant with value the function field $K(X)$.

  2. To make sense out of the definitions above one needs to show that

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

    for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional local domain ${\mathcal O}$. This will be done in the lectures.

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Your email address will not be published.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 02F5. Beware of the difference between the letter 'O' and the digit '0'.



View Remarks 110.49.2 as pdf