The Stacks project

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

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

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. $\square$