Lemma 35.35.2. Let $h : S' \to S$ be a surjective, flat morphism of schemes. The base change functor
is faithful.
Lemma 35.35.2. Let $h : S' \to S$ be a surjective, flat morphism of schemes. The base change functor
is faithful.
Proof. Let $X_1$, $X_2$ be schemes over $S$. Let $\alpha , \beta : X_2 \to X_1$ be morphisms over $S$. If $\alpha $, $\beta $ base change to the same morphism then we get a commutative diagram as follows
Hence it suffices to show that $S' \times _ S X_2 \to X_2$ is an epimorphism. As the base change of a surjective and flat morphism it is surjective and flat (see Morphisms, Lemmas 29.9.4 and 29.25.8). Hence the lemma follows from Lemma 35.35.1. $\square$
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