Lemma 26.23.2 (01L3)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.23: Monomorphisms
Lemma 26.23.2 (cite)

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A scheme morphism is a monomorphism iff its diagonal is an isomorphism.

Lemma 26.23.2. Let $j : X \to Y$ be a morphism of schemes. Then $j$ is a monomorphism if and only if the diagonal morphism $\Delta _{X/Y} : X \to X \times _ Y X$ is an isomorphism.

Proof. This is true in any category with fibre products. $\square$

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Comment #861 by Kestutis Cesnavicius on July 27, 2014 at 13:50

Suggested slogan: A scheme morphism is a monomorphism iff its diagonal is an isomorphism

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