The Stacks project

Remark 31.16.2. If $(A, \mathfrak m)$ is a Noetherian local normal domain of dimension $\geq 2$ and $U$ is the punctured spectrum of $A$, then $\Gamma (U, \mathcal{O}_ U) = A$. This algebraic version of Hartogs's theorem follows from the fact that $A = \bigcap _{\text{height}(\mathfrak p) = 1} A_\mathfrak p$ we've seen in Algebra, Lemma 10.157.6. Thus in this case $U$ cannot be affine (since it would force $\mathfrak m$ to be a point of $U$). This is often used as the starting point of the proof of Lemma 31.16.1. To reduce the case of a general Noetherian local ring to this case, we first complete (to get a Nagata local ring), then replace $A$ by $A/\mathfrak q$ for a suitable minimal prime, and then normalize. Each of these steps does not change the dimension and we obtain a contradiction. You can skip the completion step, but then the normalization in general is not a Noetherian domain. However, it is still a Krull domain of the same dimension (this is proved using Krull-Akizuki) and one can apply the same argument.


Comments (4)

Comment #2309 by on

The notion of Krull domain is not formally defined in the Stacks project at the moment.

Comment #2387 by on

Yes, you are right. I never can remember the definition. Anyway, the remark was just sketching an alternative proof of an already proven lemma, so I am going to leave it for now.

Comment #5068 by Mario Kummer on

Note that the name of the mathematician was "Hartogs" rather than "Hartog". So strictly speaking it should be "Hartogs's theorem".

There are also:

  • 2 comment(s) on Section 31.16: Complements of affine opens

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Your email address will not be published.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