The Stacks project

Lemma 35.32.2. 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 29.24.8). Hence the lemma follows from Lemma 35.32.1. $\square$


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