Let $\mathcal{F}$ be a quasi-coherent sheaf of finite type on a scheme $X$. Suppose that we have a morphism of sheaves $\varphi:\mathcal{O}_{X}^{\oplus I}\rightarrow \mathcal{F}$ such that $\varphi_{p}:\mathcal{O}_{X,p}^{\oplus I}\rightarrow \mathcal{F}_{p}$ is surjective for some $p\in X$. I want to prove that there exists an open neighborhood $U$ of $p$ such that $\varphi_{q}:\mathcal{O}_{X,q}^{\oplus I}\rightarrow \mathcal{F}_{q}$ is surjective for every $q\in U$. This is what I have tried:
Since $\mathcal{F}$ is of finite type there exist $a_{1},\ldots,a_{n}\in \mathcal{F}_{p}$ that generate $\mathcal{F}_{p}$ as $\mathcal{O}_{X,p}$-module. We may understand $a_{1},\ldots,a_{n}\in \Gamma(U,\mathcal{F})$ for some open neighborhood $U$ of $p$. I think that maybe $U$ is the open subset we are looking for. Let $b_{1},\ldots,b_{n}\in\Gamma(U,\mathcal{O}_{X}^{\oplus I})$ be such that $\varphi_{p}(U,b_{i})=(U,a_{i})$, $i\in\{1,\ldots,n\}$. I m trying to prove that $\varphi_{q}(U,b_{1}),\ldots, \varphi_{q}(U,b_{n})$ generate $\mathcal{F}_{q}$ to see that $\varphi_{q}$ is surjective, but I am not getting anywhere.
Any help would be appreciated.
These arguments are standard. It is clear that you can replace $X$ by an affine neighbourhood of $p$ and thus assume $F$ is finitely generated over $A$ where $X=\mathrm{Spec}\, A$. Then, let $G$ be the cokernel of $\phi$. This is finitely generated and $G_p=0$ by assumption. So, there exists an $f\in A$, $f(p)\neq 0$ and $G_f=0$. That is, $\phi$ is surjective on the open set $\mathrm{Spec}\,A_f$, which is a neighbourhood of $p$.