Lemma 15.11.16 (09Y6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.11: Henselian pairs
Lemma 15.11.16 (cite)

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Lemma 15.11.16. Let $(A, I)$ be a henselian pair. Let $\mathfrak p \subset A$ be a prime ideal. Then $V(\mathfrak p + I)$ is connected.

Proof. By Lemma 15.11.8 we see that $(A/\mathfrak p, I + \mathfrak p/\mathfrak p)$ is a henselian pair. Thus it suffices to prove: If $(A, I)$ is a henselian pair and $A$ is a domain, then $\mathop{\mathrm{Spec}}(A/I) = V(I)$ is connected. If not, then $A/I$ has a nontrivial idempotent by Algebra, Lemma 10.21.4. By Lemma 15.11.6 this would imply $A$ has a nontrivial idempotent. This is a contradiction. $\square$

Comments (2)

Comment #2172 by JuanPablo on August 21, 2016 at 20:47

I think here something like lemma 10.21.2 (tag 00EG) is used.

Comment #2201 by Johan on August 29, 2016 at 18:08

Yes, or the easier Lemma 10.21.4. Thanks. The corresponding change is here.

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