Lemma 51.3.1 (0BLR)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.3: Hartshorne's connectedness lemma
Lemma 51.3.1 (cite)

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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. $\square$

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