Lemma 33.22.1. Let $X \to \mathop{\mathrm{Spec}}(A)$ be a morphism of schemes. Let $A \subset A'$ be a faithfully flat ring map. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is globally generated if and only if the base change $\mathcal{F}_{A'}$ is globally generated.

Proof. More precisely, set $X_{A'} = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A')$. Let $\mathcal{F}_{A'} = p^*\mathcal{F}$ where $p : X_{A'} \to X$ is the projection. By Cohomology of Schemes, Lemma 30.5.2 we have $H^0(X_{k'}, \mathcal{F}_{A'}) = H^0(X, \mathcal{F}) \otimes _ A A'$. Thus if $s_ i$, $i \in I$ are generators for $H^0(X, \mathcal{F})$ as an $A$-module, then their images in $H^0(X_{A'}, \mathcal{F}_{A'})$ are generators for $H^0(X_{A'}, \mathcal{F}_{A'})$ as an $A'$-module. Thus we have to show that the map $\alpha : \bigoplus _{i \in I} \mathcal{O}_ X \to \mathcal{F}$, $(f_ i) \mapsto \sum f_ i s_ i$ is surjective if and only if $p^*\alpha$ is surjective. This we may check over an affine open $U = \mathop{\mathrm{Spec}}(B)$ of $X$. Then $\mathcal{F}|_ U$ corresponds to a $B$-module $M$ and $s_ i|_ U$ to elements $x_ i \in M$. Thus we have to show that $\bigoplus _{i \in I} B \to M$ is surjective if and only if the base change $\bigoplus _{i \in I} B \otimes _ A A' \to M \otimes _ A A'$ is surjective. This is true because $A \to A'$ is faithfully flat. $\square$

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