Lemma 35.35.3 (0240)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.35: Fully faithfulness of the pullback functors
Lemma 35.35.3 (cite)

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Lemma 35.35.3. In the situation of Lemma 35.34.6 assume that $f : X' \to X$ is surjective and flat. Then the pullback functor is faithful.

Proof. Let $(V_ i, \varphi _ i)$, $i = 1, 2$ be descent data for $X \to S$. Let $\alpha , \beta : V_1 \to V_2$ be morphisms of descent data. Suppose that $f^*\alpha = f^*\beta $. Our task is to show that $\alpha = \beta $. Note that $\alpha $, $\beta $ are morphisms of schemes over $X$, and that $f^*\alpha $, $f^*\beta $ are simply the base changes of $\alpha $, $\beta $ to morphisms over $X'$. Hence the lemma follows from Lemma 35.35.2. $\square$

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