Lemma 59.31.5 (04FU)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.31: Supports of abelian sheaves
Lemma 59.31.5 (cite)

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Lemma 59.31.5. The support of a sheaf of rings on $S_{\acute{e}tale}$ is closed.

Proof. This is true because (according to our conventions) a ring is $0$ if and only if $1 = 0$, and hence the support of a sheaf of rings is the support of the unit section. $\square$

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