This is my attempt to solve Hartshorne exercise 5.7 section II.
Let $X$ be a noetherian scheme and $\mathcal{F}$ be a coherent sheaf. I want to prove the following:
If $\mathcal{F}_x$ is a free $\mathcal{O}_{X,x}$-module for some $x\in X$ then there exists a neighbourhood $U$ of $x$ such that $\mathcal{F}_{|U}$ is free
My attempt
Let $x_0$ be the point of $X$ in which $\mathcal{F}$ has free stalk. Since $\mathcal{F}$ is coherent then it is of finite type, so let $s_1,\dots,s_n\in\mathcal{F}(U)$ be local sections in a neighbourhood of $x_0$ such that $s_{1,x_0},\dots,s_{n,x_0}$ generate $\mathcal{F}_{x_0}\cong\mathcal{O}^n_{X,x_0}$. One can easily show that $s_{1,x},\dots,s_{n,x}$ generate $\mathcal{F}_{x}$ for all $x\in V\subset U$. In particular we obtain a surjective morphism \begin{equation*}\phi:\mathcal{O}^n_{|V}\to\mathcal{F}_{|V} \end{equation*} But $\mathcal{F}$ is coherent so the kernel of such morphism is finitely generated and it is zero over $x_0$ ( I mean the stalk of the sheaf kernel is zero at $x_0$). In particular the induced map $\phi_{x_0}$ on the stalk is an isomorphism. So by some lemmata on stack exchange it follows that $\phi$ is injective when restricted to some $W\subset V$ neighbourhood of $x_0$ and in particular it gives an isomorphism between $\mathcal{O}^n_{|W}$ and $\mathcal{F}_{|W}$.
Now, I have two questions:
1) Does my argument work?
2) Where did I use the noetherian hypothesis on $X$? Am I missing something?