Lemma 17.11.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation.

1. If $\psi : \mathcal{O}_ X^{\oplus r} \to \mathcal{F}$ is a surjection, then $\mathop{\mathrm{Ker}}(\psi )$ is of finite type.

2. If $\theta : \mathcal{G} \to \mathcal{F}$ is surjective with $\mathcal{G}$ of finite type, then $\mathop{\mathrm{Ker}}(\theta )$ is of finite type.

Proof. Proof of (1). Let $x \in X$. Choose an open neighbourhood $U \subset X$ of $x$ such that there exists a presentation

$\mathcal{O}_ U^{\oplus m} \xrightarrow {\chi } \mathcal{O}_ U^{\oplus n} \xrightarrow {\varphi } \mathcal{F}|_ U \to 0.$

Let $e_ k$ be the section generating the $k$th factor of $\mathcal{O}_ X^{\oplus r}$. For every $k = 1, \ldots , r$ we can, after shrinking $U$ to a small neighbourhood of $x$, lift $\psi (e_ k)$ to a section $\tilde e_ k$ of $\mathcal{O}_ U^{\oplus n}$ over $U$. This gives a morphism of sheaves $\alpha : \mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus n}$ such that $\varphi \circ \alpha = \psi$. Similarly, after shrinking $U$, we can find a morphism $\beta : \mathcal{O}_ U^{\oplus n} \to \mathcal{O}_ U^{\oplus r}$ such that $\psi \circ \beta = \varphi$. Then the map

$\mathcal{O}_ U^{\oplus m} \oplus \mathcal{O}_ U^{\oplus r} \xrightarrow {\beta \circ \chi , 1 - \beta \circ \alpha } \mathcal{O}_ U^{\oplus r}$

is a surjection onto the kernel of $\psi$.

To prove (2) we may locally choose a surjection $\eta : \mathcal{O}_ X^{\oplus r} \to \mathcal{G}$. By part (1) we see $\mathop{\mathrm{Ker}}(\theta \circ \eta )$ is of finite type. Since $\mathop{\mathrm{Ker}}(\theta ) = \eta (\mathop{\mathrm{Ker}}(\theta \circ \eta ))$ we win. $\square$

Comment #6198 by Masa on

In the second sentence of Lemma 17.11.3, "a $\mathcal O_X$-module" should be "an $\mathcal O_X$-module".

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