Lemma 35.25.1 (06NC)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.25: Application of fpqc descent of properties of morphisms
Lemma 35.25.1 (cite)

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Lemma 35.25.1. Let $f : X \to Y$ be a flat, quasi-compact, surjective monomorphism. Then f is an isomorphism.

Proof. As $f$ is a flat, quasi-compact, surjective morphism we see $\{ X \to Y\}$ is an fpqc covering of $Y$ . The diagonal $\Delta : X \to X \times _ Y X$ is an isomorphism (Schemes, Lemma 26.23.2). This implies that the base change of $f$ by $f$ is an isomorphism. Hence we see $f$ is an isomorphism by Lemma 35.23.17. $\square$

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