The Stacks project

Lemma 35.7.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of locally ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. If

  1. $f$ is open as a map of topological spaces,

  2. $f$ is surjective and flat, and

  3. $f^*\mathcal{F}$ is of finite type,

then $\mathcal{F}$ is of finite type.

Proof. Let $y \in Y$ be a point. Choose a point $x \in X$ mapping to $y$. Choose an open $x \in U \subset X$ and elements $s_1, \ldots , s_ n$ of $f^*\mathcal{F}(U)$ which generate $f^*\mathcal{F}$ over $U$. Since $f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{O}_ X$ we can after shrinking $U$ assume $s_ i = \sum t_{ij} \otimes a_{ij}$ with $t_{ij} \in f^{-1}\mathcal{F}(U)$ and $a_{ij} \in \mathcal{O}_ X(U)$. After shrinking $U$ further we may assume that $t_{ij}$ comes from a section $s_{ij} \in \mathcal{F}(V)$ for some $V \subset Y$ open with $f(U) \subset V$. Let $N$ be the number of sections $s_{ij}$ and consider the map

\[ \sigma = (s_{ij}) : \mathcal{O}_ V^{\oplus N} \to \mathcal{F}|_ V \]

By our choice of the sections we see that $f^*\sigma |_ U$ is surjective. Hence for every $u \in U$ the map

\[ \sigma _{f(u)} \otimes _{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} : \mathcal{O}_{X, u}^{\oplus N} \longrightarrow \mathcal{F}_{f(u)} \otimes _{\mathcal{O}_{Y, f(u)}} \mathcal{O}_{X, u} \]

is surjective. As $f$ is flat, the local ring map $\mathcal{O}_{Y, f(u)} \to \mathcal{O}_{X, u}$ is flat, hence faithfully flat (Algebra, Lemma 10.39.17). Hence $\sigma _{f(u)}$ is surjective. Since $f$ is open, $f(U)$ is an open neighbourhood of $y$ and the proof is done. $\square$

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