Hartshorne's connectedness

[Proposition 2.1, Hartshorne-connectedness]

Lemma 51.3.1. Let $A$ be a Noetherian local ring of depth $\geq 2$. Then the punctured spectra of $A$, $A^ h$, and $A^{sh}$ are connected.

Proof. Let $U$ be the punctured spectrum of $A$. If $U$ is disconnected then we see that $\Gamma (U, \mathcal{O}_ U)$ has a nontrivial idempotent. But $A$, being local, does not have a nontrivial idempotent. Hence $A \to \Gamma (U, \mathcal{O}_ U)$ is not an isomorphism. By Lemma 51.2.2 we conclude that either $H^0_\mathfrak m(A)$ or $H^1_\mathfrak m(A)$ is nonzero. Thus $\text{depth}(A) \leq 1$ by Dualizing Complexes, Lemma 47.11.1. To see the result for $A^ h$ and $A^{sh}$ use More on Algebra, Lemma 15.45.8. $\square$

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Your email address will not be published.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