The Stacks project

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 15.44.8. $\square$


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