The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B57. Beware of the difference between the letter 'O' and the digit '0'.