Lemma 35.35.2. Let h : S' \to S be a surjective, flat morphism of schemes. The base change functor
\mathit{Sch}/S \longrightarrow \mathit{Sch}/S', \quad X \longmapsto S' \times _ S X
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
\xymatrix{ X_2 \ar[d]^\alpha & S' \times _ S X_2 \ar[l] \ar[d] \ar[r] & X_2 \ar[d]^\beta \\ X_1 & S' \times _ S X_1 \ar[l] \ar[r] & X_1 }
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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